University  of  California. 


ELEMENTARY  TREATISE 


•    ON 

ASTRONOMY: 

*  <** 
IN  FOUK  PAKTS. 

• 

OONTAINIM.G 

i  SYSTEMATIC  AND  COMPEEHENSIVE  EXPOSITION  OF  THE  THEORY, 
AND  THE  MOKE  IMPORTANT  PRACTICAL  PROBLEMS ; 

WITH 

SOLAR,  LUNAR,  AND  OTHER  ASTRONOMICAL  TABLES, 

DESIGNED  FOB  USE  AS  A 

TEXT-BOOK  IN  COLLEGES  AND  THE  HIGHER  ACADEMIES. 


- 

BY 

WILLIAM  A.  NOBTON,  A.  M,, 

FELLOW  OF  THE  AMERICAN  PHILOSOPHICAL  SOCIETY,  AND  OF  THE  AMERICAN 

ACADEMY  OF  ARTS  AND  SCIENCES,  AND  CORRESPOND****  -—:=±i==^z 
MEMBER  OF  THE  NATIONAL  INSTITUTE. 

LIBKA  i; 

. 

THIBD  EDITION.  Xl  Y  |f^  I  T  V 

CORRECTED,  IMPROVED,  AND  ENLARGED.  (    '    \     t  »  V    ! 

NEW    YORK: 
JOHN    WILEY,    167    BROADWAY. 

.-   *•;..  v  i' "          '   

1853. 


Entered  according  to  Act  of  Congress,  in  the  year  1845. 

By  WILLIAM  A  NORTON, 
In  the  Clerk's  Office  of  the  District  Court  for  the  Southern  District  of  New  York. 


Stereotyped  by 
RICHARD  C.  VALENTINE. 

17  Dnteh-itreet,  New  YorK. 


PREFACE 

TO    THE    FIRST    EDITION. 


THE  object  for  which  the  present  treatise  on  Astronomy  has 
been  written,  is  to  provide  a  suitable  text-book  for  the  use  of  the 
students  of  Colleges  and  the  higher  Academies,  and  at  the  same 
time  to  furnish  the  practical  astronomer  with  rules,  or  formulae, 
and  accurate  tables  for  performing  the  more  important  astronomi- 
cal calculations. 

It  is  divided  into  four  Parts.  The  first  three  Parts  contain  the 
theory  :  the  First  Part  treating  of  the  determination  of  the  places 
and  motions  of  the  heavenly  bodies  ;  the  Second,  of  the  phenom- 
ena resulting  from  the  motions  of  these  bodies,  and  of  their  ap- 
pearances, dimensions,  and  physical  constitution ;  and  the  Third, 
of  the  theory  of  Universal  Gravitation.  The  Fourth  Part  consists 
of  Practical  Problems,  which  are  solved  with  the  aid  of  the  Tables 
appended  to  the  work.  An  Appendix  is  added,  containing  a  large 
collection  of  useful  trigonometrical  formulae,  and  such  investiga- 
tions of  astronomical  formulae  as,  from  their  length,  could  not, 
consistently  with  the  plan  of  the  work,  be  admitted  into  the  text, 
and  which  it  was  still  deemed  advisable  to  retain,  for  the  benefit 
of  the  few  who  might  wish  to  pursue  them. 

The  chief  peculiarities  of  this  treatise,  as  compared  with  the 
kindred  works  now  in  use  in  our  Colleges,  are, — 1.  The  adoption 
of  the  Copernican  System  as  an  hypothesis  at  the  outset,  leaving 
it  to  be  established  by  the  agreement  between  the  conclusions  to 
which  it  leads  and  the  results  of  observation.  2.  A  connected  ex- 
position of  the  principles  and  methods  of  astronomical  observation, 
embracing  the  doctrine  of  the  sphere,  the  construction  and  use  of 


.:*/ 

Iv  PREFACE    TO    THE    FIRST   EDITION. 

the  principal  astronomical  instruments,  and  the  theory  of  the  cor- 
rections for  refraction,  parallax,  aberration,  precession,  and  nuta- 
tion. 3.  The  exhibition  of  the  methods  of  determining  the  motions 
and  places  of  the  different  classes  of  the  heavenly  bodies,  in  one 
connection.  4.  The  explanation  of  the  principles  of  the  construc- 
tion of  astronomical  tables.  5.  The  addition  of  a  chapter  on  the 
measurement  of  time,  embracing  the  explanation  of  the  different 
kinds  of  time,  the  processes  by  which  one  is  converted  into  an- 
other, the  methods  of  determining  the  time  from  astronomical  obser- 
vations with  the  transit  instrument  and  sextant,  and  the  calendar. 
6.  The  contemplation  of  the  phenomena  of  the  aspect  and  appa- 
rent motion  of  the  heavenly  bodies  as  consequences  of  their  motions 
in  space,  and  the  deduction  of  the  various  circumstances  of  these 
phenomena  from  the  theory  of  the  orbitual  motions  previously  es- 
tablished. 7.  A  comprehensive  view  of  the  theory  of  Universal 
Gravitation,  followed  out  into  its  various  consequences.  8.  An 
exposition  of  the  operations  of  the  disturbing  forces  in  producing 
the  principal  perturbations  of  the  motions  of  the  Solar  System. 
9.  The  solution  of  Practical  Problems  by  means  of  logarithmic 
formulae  instead  of  rules.  10.  The  addition  of  lunar,  solar,  and 
other  astronomical  tables,  of  peculiar  accuracy  and  improved  ar- 
rangement. 

It  may  further  be  mentioned,  that  many  of  the  investigations 
have  been  materially  simplified,  and  that  the  aim  has  been  to  in- 
troduce into  all  of  them  as  much  simplicity  and  uniformity  of 
method  as  possible.  Particular  attention  has  also  been  paid  to 
the  diagrams,  it  being  of  great  importance  that  they  should  convey 
correct  notions  to  the  mind  of  the  student. 

The  Problems  in  the  Fourth  Part  /are  principally  for  making 
calculations  relative  to  the  Sun,  Moon,  and  Fixed  Stars.  The 
Tables  of  the  Sun  and  Moon,  used  in  finding  the  places  of  these 
bodies,  have,  for  the  most  part,  been  abridged  and  computed  from 
the  tables  of  Delambre,-  as  corrected  by  Bessel,  and  those  of 
Burckhardt ;  and  the  Tables  of  Epochs  have  all  been  reduced  to 
the  meridian  of  Greenwich.  These  Tables  will  give  the  places 


PREFACE    TO    THE    FIRST   EDITION.  V 

and  motions  of  the  Sun  and  Moon  within  a  fraction  of  a  second  of 
the  tables  from  which  they  were  derived.  But  as  this  degree  of 
accuracy  will  not  generally  be  required,  rules  are  also  given  in 
the  Fourth  Part  for  obtaining  approximate  results.  The  entire  set 
of  Tables  has  been  stereotyped,  and  great  pains  has  been  taken, 
by  repeated  revisions  and  verifications,  to  render  them  accurate. 

The  principal  astronomical  works  which  have  been  consulted, 
are  those  of  Vince,  Gregory,  Woodhouse,  Delambre,  Biot,  La- 
place, Herschel,  and  Gummere ;  also  FranccBur's  Uranography, 
Francmur's  Practical  Astronomy,  Encyclopedia  Metropolitana, 
Article  "  Astronomy,"  and  Baily's  Tables  and  Formula.  Free 
use  has  been  made  of  the  methods  of  investigation  and  demonstra- 
tion pursued  in  these  treatises,  such  modifications  being  intro- 
duced, in  those  which  have  been  adopted,  as  the  plan  of  the  work 
required. 

New  Yo'*,  January,  1839. 


PREFACE 

TO    THE    SECOND    EDITION. 


IN  preparing  a  new  edition  of  the  present  treatise,  material  al- 
terations, and,  it  is  hoped,  improvements  have  been  made  in  it. 
The  more  abstruse  parts  are  now  printed  in  smaller  type,  and 
their  connection  with  the  other  portions  of  the  book  is  made  such 
that  they  can  be  pursued  or  omitted  at  pleasure :  by  which  the 
opportunity  is  afforded  of  making  a  selection  between  two  courses 
of  study,  differing  materially  in  extent,  and  in  the  amount  of  labor 
and  mathematical  attainment  required  for  their  acquisition.  Wood- 
cuts have  also  been  substituted  for  the  original  plates,  as  more 
convenient  to  the  student ;  and  for  the  sake  of  more  ample  illus- 
tration, nearly  fifty  new  diagrams  have  been  added.  Many  of 
these  are  illustrative  of  the  telescopic  appearances  of  the  planets 
and  other  heavenly  bodies.  Considerable  additions  have  been 
made  to  several  of  the  Chapters ;  especially  to  the  Chapter  on 
Instruments,  and  those  in  which  the  appearances  and  physical 
constitution  of  the  heavenly  bodies  are  treated  of.  These  are,  for 
the  most  part,  printed  in  a  small-sized  type,  as  well  as  the  parts 
above  specified.  The  Chapters  on  Comets  have  been  rewritten. 
The  Author  has  also  endeavored,  in  many  instances  which  need 
not  be  enumerated,  to  profit  by  such  criticisms  and  suggestions  of 
improvement  as  have  been  made  by  others,  as  well  as  by  his  own 
experience  in  the  use  of  the  work  as  a  text-book. 

The  Tables  remain  unaltered  ;  with  the  exception  of  Tables  L, 
I'L,  III.,  and  IV.,  which  have  been  rendered  more  accurate.  Fre- 
quent comparisons,  since  the  publication  of  the  first  edition,  of  the 
Lunar  and  Solar  Tables  with  the  places  of  the  Moon  and  Sun,  as 


PREFACE    TO    THE    SECOND   EDITION.  vii 

given  in  the  Nautical  Almanac  and  the  Connaissance  des  Terns, 
have  furnished  additional  confirmation  of  their  accuracy. 

Notwithstanding  the  considerable  augmentation  which  the  work 
has  received,  the  retail  price  of  it  is  very  much  reduced. 

The  references  in  the  text  to  the  investigations  of  astronomical 
formulae  in  the  Appendix,  were  omitted,  in  preparing  this  edition, 
under  the  expectation  that  the  new  matter  to  be  inserted  would 
render  the  omission  of  these  investigations  necessary.  They  are, 
however,  retained ;  and  the  articles  are  designated  in  which  men- 
tion is  made  of  such  formulas. 

In  addition  to  the  Astronomical  works  mentioned  in  the  preface 
to  the  first  edition,  the  Author  has  particularly  consulted,  in  the 
preparation  of  this  edition,  besides  periodicals,  Littrow's  Wonders 
of  the  Heavens,  KendaWs  Uranography,  NichoVs  Phenomena  of 
the  Solar  System,  Nicholas  Architecture  of  the  Heavens,  and  Ma- 
son's Introduction  to  Practical  Astronomy. ,  His  acknowledgments 
are  due  to  Professor  Kendall  for  the  copy  which  he  was  permitted 
to  take  of  the  delineation  of  the  great  comet  of  1843,  given  in  his 
Uranography. 

Where  passages  have  been  borrowed  entire  from  any  author, 
credit  has  been  given  in  the  usual  way,  viz.,  by  references  to 
specifications  of  title,  &c.,  inserted  at  the  bottom  of  the  page. 
To  these  it  should  be  added  that  the  greater  portion  of  the  Chap- 
ter on  the  Calendar,  after  the  first  paragraph,  is  taken  from  Wood- 
house's  Astronomy,  and  most  of  Art.  463,  from  Gregory's  Astron- 
omy. Particular  assistance  has  also  been  derived,  in  Part  IV., 
from  Gummere's  Astronomy.  It  would  be  idle  in  every  new 
scientific  treatise,  to  attempt  to  designate  all  the  instances  in  which 
the  same  forms  of  expression  and  the  same  methods  of  investiga- 
tion may  have  been  adopted,  that  occur  in  other  kindred  treatises. 
DELAWARE  COLLEGE, 

Newark,  Del,  June,  1845. 


PREFACE 

TO    THE    THIRD    EDITION. 


SINCE  the  publication  of  the  previous  edition,  numerous  im- 
portant and  highly  interesting  astronomical  discoveries  have 
been  made.  These  have  been  introduced  into  the  present 
edition,  by  appending*a  collection  of  Notes  to  the  text.  The 
references  to  these  notes,  inserted  in  the  text,  will  bring  the 
different  topics  of  which  they  treat  to  the  notice  of  the  stu- 
dent, in  the  proper  connection,  while  they  will  collectively 
form  a  brief  exposition  of  the  progress  recently  made  in  astro- 
nomical science.  It  has  been  the  intention  to  make  this  edi- 
tion a  faithful  picture  of  the  present  state  of  the  science ;  in 
so  far  as  this  end  could  be  attained  within  the  limits  which 
should  be  observed  in  the  preparation  of  a  college  text-book. 

PROVIDENCE,  April,  1852. 


TABLE    OF    CONTENTS. 


INTRODUCTION. 
General  Notions — General  Phenomena  of  the  Heavens 

PART  I 

ON  THE  DETERMINATION  OF  THE  PLACES  AND  MOTIONS  OF 
THE  HEAVENLY  BODIES. 

CHAPTER  I. 

On  the  Celestial  and  Terrestrial  Spheres         .' ;   .      °.  ^    V       .        n 

CHAPTER  II. 

On  the  Construction  and  Use  of  the  Principal  Astronomical  Instruments  23 

Transit  Instrument          -        -        -        -.-.,*        .  26 

Astronomical  Clock      ^  .-f  ,,<:^%..-.* :    /-^     •      ,•*  31 

Astronomical  Circle         -        -        -       V'_.  ->  \' '  » '  '    '*        -  32 

Altitude  and  Azimuth  Instrument     » ,      *,       -        -        -        -  3$ 

Equatorial      -  ",   - ,..,,V*  /.', "- /-;y-^t •;•'&.  37 

Sextant          >        .        .        .        4     '.!•,       ....  39 

Errors  of  Instrumental  Admeasurement    -----  43 

Telescope       -        -        ....        .        .       .       -  #. 

CHAPTER  III. 

On  the  Corrections  of  the  Co-ordinates  of  the  Observed  Place  of  a 

Heavenly  Body          -        -        -        -        -        .  ,^    . „       .  43 

Refraction      -        -        -        -        -        *        -        *       ,„      :-  44 

/Parallax          ....        j--'      .;        „        .        .        .  49 

f  Aberration      --.-.-        .    •  >.        -        -  55 

f  Precession  and  Nutation      ^4>^i.^-%-      *'      ...  60 
Remarks  on  the  Corrections. — Verification   of  the  Hypothesis 

that  the  Diurnal  Motion  of  the  Stars  is  Uniform  and  Circular  66 

2 


TABLE    OF    CONTENTS. 


CHAPTER  IV. 

PACK 

Of  the  Earth— its  Figure  and  Dimensions — Latitude  and  Longitude 
of  a  Place  ..--------66 

Determination  of  the  Latitude  and  Longitude  of  a  Place  69 

CHAPTER  V. 
Of  the  Places  of  the  Fixed  Stars 72 

CHAPTER  VI. 
Of  the  apparent  motion  of  the  Sun  in  the  Heavens  -  77 

CHAPTER  VII. 

Of  the  Motions  of  the  Sun,  Moon,  and  Planets,  in  their  orbits            -  82 

Kepler's  Laws        -                           - ;    '••'•••  '< "--;,      ,-        -        -  ib. 

Definitions  of  Terms        -        -        -        --        -         -        -  85 

Elements  of  the  Orbit  of  a  Planet    -        -        -        -  .      -        -  87 

Methods  of  Determining  the  Elements  of  the  Sun's  Apparent 

Orbit,  or  of  the  Earth's  Real  Orbit 88 

Methods  of  Determining  the  Elements  of  the  Moon's  Orbit        -  92 

Methods  of  Determining  the  Elements  of  a  Planet's  Orbit          -  94 

Mean  Elements  and  their  Variations         -        -        -        -        -  102 

CHAPTER  VIII. 

On  the  Determination  of  the  Place  of  a  Planet,  or  of  the  Sun,  or 
Moon,  for  a  Given  Time,  by  the  Elliptical  Theory ;  and  of  the 

Verification  of  Kepler's  Laws           •' 105 

Place  of  a  Planet,  or  of  the  Sun  or  Moon  in  its  Orbit       -        -  ib. 

Heliocentric  Place  of  a  Planet        -        -        -        .   -    .    .  V  106 

Geocentric  Place  of  a  Planet          -                 -  107 

Places  of  the  Sun  and  Moon           -        -        -        .,-  108 

Verification  of  Kepler's  Laws                                   *             -;  "£-?  ib. 

CHAPTER  IX. 

On  the  Inequalities  of  the  Motions  of  the  Planets  and  of  the  Moon ; 
and  of  the  Construction  of  Tables  for  finding  the  Places  of  these 

Bodies 109 

Construction  of  Tables  -        .        _        .        .        .        .        114 


TABLE    OF    CONTENTS. 


CHAPTER  X. 

PAGE 

Of  the  Motions  of  the  Comets       *V,«'  .  -        -        •        -        117 


CHAPTER  XL 

Of  the  Motions  of  the  Satellites    -        -        -        ....  134 

CHAPTER  XII. 

the  Measurement  of  Time        -        -        -        -        -        -        -  137 

Different  Kinds  of  Time        ----„,.    ••  •  «,.      _«.  -  -  ib. 

Conversion  of  one  Species  of  Time  into  another      -        -        -  128 
Determination  of  the  Time  and  Regulation  of  Clocks  by  Astro- 

nomical Observations          -        -        -        -        -        .        -  130 

^f  the  Calendar    -        -      v,,  ^  ,^  :,   ^  ,    ^    •,;....      .        .  133 


PART  II. 

ON  THE  PHENOMENA  RESULTING  FROM  THE  MOTIONS  OF 
THE  HEAVENLY  BODIES,  AND  ON  THEIR  APPEARANCES, 
DIMENSIONS,  AND  PHYSICAL  CONSTITUTION. 

CHAPTER  XIII. 

Of  the  Sun  and  the  Phenomena  attending  its  Apparent  Motions       -  137 

Inequality  of  Days         -        -        -*  •     .«•"       -  ib» 

Twilight        -        -        -       V        V 141 

The  Seasons      *  -     J^        -    ,    - 144 

Appearance,  Dimensions,  and  Physical  Constitution  of  the  Sun  147 

CHAPTER  XIV. 

Of  the  Moon  and  its  Phenomena             -        -        -        -        ••*>.*  153 

Phases  of  the  Moon       -        *        *        *        *        *        •        *  ** 

Moon's  Rising,  Setting,  and  Passage  over  the  Meridian           -  .  155 

Rotation  and  Librations  of  the  Moon       -  .    .•%-      -    '   —        -  158 

Dimensions  and  Physical  Constitution  of  the  Moon          *        -  159 


Hi  TABLE    OF    CONTENTS. 


CHAPTER  XV. 

PAS« 

Eclipses  of  the  Sun  and  Moon — Occultations  of  the  Fixed  Stars     -  162 

Eclipses  of  the  Moon    -        -    r  *.       -        ....  ib. 

Eclipses  of  the  Sun 171 

Occultations          -        -  •'"»  -,: '.'» 183 


CHAPTER  XVI. 

Of  the  Planets  and  the  Phenomena  occasioned  by  their  Motions  in 

Space *r    '-  184 

Apparent  Motions  of  the  Planets  with  respect  to  the  Sun  ib. 

Stations  and  Retrogradations  of  the  Planets     -        -        «•        --  187 

Phases  of  the  Inferior  Planets        ....        W:    : 4'  190 

Transits  of  the  Inferior  Planets       -        -         -         -        *        -  191 
Appearances,  Dimensions,  Rotation,  and  Physical  Constitution 

of  the  Planets             192 


CHAPTER  XVII. 

Of  Comets                                     -        -        -        -        -  '     -        -  201 

Their  General  Appearance — Varieties  of  Appearance      -        -  ib. 

Form,  Structure,  and  Dimensions  of  Comets        Tv-        *        -  205 

Physical  Constitution  of  Comets    -        -        -        .       *        .  207 

CHAPTER  XVIII. 

Of  the  Fixed  Stars       *,.     *; 211 

Their  Number  and  Distribution  over  the  Heavens           -•;       -  ib. 

Annual  Parallax  and  Distance  of  the  Stars      ....  213 

Nature  and  Magnitude  of  the  Stars         .....  216 

Variable  Stars -        -        .  917 

Double  Stars       V ^                 -  219 

Proper  Motions  of  the  Stars           -        *';.-»        .        -  222 

Clusters  of  Stars — Nebulae 223 

Distance  and  Magnitude  of  Nebulje         .....  227 

Structure  of  the  Material  Universe— Nebular  Hypothesis        -  229 


TABLE    OF    CONTENTS.  xiii 

PART    III. 

OF  THE  THEORY  OF  UNIVERSAL  GRAVITATION. 
CHAPTER  XIX. 

ffMP 
Of  the  Principle  of  Universal  Gravitation      -V,^\y       -        -        231 

CHAPTER  XX. 

Theory  of  the  Elliptic  Motion  of  the  Planets  -     '"'  *        234 

CHAPTER  XXI. 

Theory  of  the  Perturbations  of  the  Elliptic  Motion  of  the  Planets 

and  of  the  Moon          ^--...-^      ......        239 

CHAPTER  XXII. 

Of  the  Relative  Masses  and  Densities  of  the  Sun,  Moon,  and  Planets ; 

and  of  the  Relative  Intensity  of  the  Gravity  at  their  surface     -        249 

CHAPTER  XXIII. 

Of  the  Figure  and  Rotation  of  the  Earth ;  and  of  the  Precession  of 

the  Equinoxes  and  Nutation       *?  ».      .•>-'      - J'.    •  ",-'"  ••'•»'     *        251 

CHAPTER  XXIV. 

the  Tides        *' '"'  '"--'., •   ,  '-;      i."     -  ' ..*  .'•    .-        -        -        254 


PART  IV. 
ASTRONOMICAL  PROBLEMS. 

EXPLANATIONS  OF  THE  TABLES      -        -        f '^  t        "    '"    "        "        ^^ 

PROB.  I.  To  work,  by  logistical  logarithms,  a  proportion  the  terms 
of  which,  are  degrees  and  minutes,  or  minutes  and  seconds,  of 
an  arc  ;  or  hours  and  minutes,  or  minutes  and  seconds,  of  time  266 


fclV  TABLE    OF    CONTENTS. 

PAGI 

PBOB.  II.  To  take  from  a  table  the  quantity  corresponding  to  a 
given  value  of  the  argument,  or  to  given  values  of  the  arguments 

of  the  table -        '        '  *     '        '        267 

PROB.  III.     To  convert  Degrees,  Minutes,  and  Seconds  of  the  Equa- 
tor into  Hours,  Minutes,  &c.,  of  Time          v?        -        -       '-        273 
PROB.  IV.     To  convert  Time  into  Degrees,  Minutes,  and  Seconds  ib. 

PROB.  V.     The  Longitudes  of  two  Places,  and  the  Time  at  one  of 

them  being  given,  to  find  the  corresponding  time  at  the  other  274 

PROB.  VI.  The  Apparent  time  being  given,  to  find  the  correspond- 
ing Mean  Time ;  or,  the  Mean  Time  being  given,  to  find  the 
Apparent  -  ,r  -  -  -  -  '  ^75 

PROB.  VII.  To  correct  the  Observed  Altitude  of  a  Heavenly  Body 

for  Refraction  -  278 

PROB.  VIII.  The  Apparent  Altitude  of  a  Heavenly  Body  being 

given,  to  find  its  True  Altitude  -  -  -  -  -  -  279 

PROB.  IX.  To  find  the  Sun's  Longitude,  Hourly  Motion,  and  Semi- 
diameter,  for  a  given  Time,  from  the  Tables  ...  281 

PROB.  X.     To  find  the  Apparent  Obliquity  of  the  Ecliptic,  for  a 

given  Time,  from  the  Tables  -        «•>•'''  .*  ; ./< •- •'£  '-*<•*.  \-  ^        283 

PROB.  XL     Given  the  Sun's  Longitude  and  the  Obliquity  of  the 

Ecliptic,  to  find  his  right  Ascension  and  Declination        -         -        284 

PROB.  XII.     Given  the  Sun's  Right  Ascension  and  the  Obliquity  of 

the  Ecliptic,  to  find  his  Longitude  and  Declination        -jgfcr ,    .^        285 

PROB.  XIII.  The  Sun's  Longitude  and  the  Obliquity  of  the  Ecliptic- 
being  given,  to  find  the  Angle  of  Position  -  -  -  ,  .  •  _  285 

PROB.  XIV.  To  find  from  the  Tables,  the  Moon's  Longitude,  Lati- 
tude, Equatorial  Parallax,  Semi-diameter,  and  Hourly  Motions 
in  Longitude  and  Latitude,  for  a  given  Time  ...  286 

PROB.  XV.     The  Moon's  Equatorial  Parallax,  and  the  Latitude  of 

a  Place,  being  given,  to  find  the  Reduced  Parallax  and  Latitude        295 

PROB.  XVI.     To  find  the  Longitude  and  Altitude  of  the  Nonagesi- 

mal  Degree  of  the  Ecliptic,  for  a  given  Time  and  Place  -          ib. 

PROB.  XVII.  To  find  the  Apparent  Longitude  and  Latitude,  as 
affected  by  Parallax,  and  the  Augmented  Semi-diameter  of  the 
Moon ;  the  Moon's  True  Longitude,  Latitude,  Horizontal  Semi- 
diameter,  and  Equatorial  Parallax,  and  the  Longitude  and  Alti- 
tude of  the  Nonagesimal  Degree  of  the  Ecliptic,  being  given  298 

PBOB.  XVIII.  To  find  the  Mean  Right  Ascension  and  Declination, 
or  Longitude  and  Latitude  of  a  Star,  for  a  given  Time,  from  the 
Tables 302 


TABLE    OP    CONTENTS.  XV 

PAOB 
PROB.  XIX.     To  find  the  Aberrations  of  a  Star  in  Right  Ascension 

and  Declination,  for  a  given  Day  -        ....        393 

PROB.  XX.     To  find  the  Nutations  of  a  Star  in  Right  Ascension 

and  Declination,  for  a  given  Day  -        -        -        -        -        304 

PROB.  XXI.  To  find  the  Apparent  Right  Ascension  and  Declina- 
tion of  a  Star,  for  a  given  day  ......  306 

PROB.  XXII.     To  find  the  Aberrations  of  a  Star  in  Longitude  and 

Latitude,  for  a  given  Day -  307 

PROB.  XXIII.     To  find  the  Apparent  Longitude  and  Latitude  of  a 

Star,  for  a  given  Day  -  ib. 

PROB.  XXIV.  To  compute  the  Longitude  and  Latitude  of  a 
Heavenly  Body  from  its  Right  Ascension  and  Declination,  the 
Obliquity  of  the  Ecliptic  being  given  -----  308 

PROB.  XXV.  To  compute  the  Right  Ascension  and  Declination 
of  a  Heavenly  Body  from  its  Longitude  and  Latitude,  the 
Obliquity  of  the  Ecliptic  being  given  -  309 

PROB.  XXVI.  The  Longitude  and  Declination  of  a  Body  being 
given,  and  also  the  Obliquity  of  the  Ecliptic,  to  find  the  Angle 
ofPosition  310 

PROB.  XXVII.     To  find  from  the  Tables  the  Time  of  New  or  Full 

Moon,  for  a  given  Year  and  Month         -        -        -        -        -        311 

PROB.  XXVIII.  To  determine  the  number  of  Eclipses  of  the  Sun 
and  Moon  that  may  be  expected  to  occur  in  any  given  Year, 
and  the  Times  nearly  at  which  they  will  take  place  -  -  314 

PROB.  XXIX.     To  calculate  an  Eclipse  of  the  Moon  317 

PROB.  XXX.     To  calculate  an  Eclipse  of  the  Sun,  for  a  given  Place        321 

PROB.  XXXI.  To  find  the  Moon's  Longitude,  &c.,  from  the  Nau- 
tical Almanac  ---------  338 


APPENDIX. 

TRIGONOMETRICAL  FORMULAE 

I.  Relative  to  a  Single  arc  or  angle  a  ib. 

II.  Relative  to  Two  Arcs  a  and  b,  of  which  a  is  supposed  to  be 

the  greater         - #• 

III.  Trigonometrical  Series 

IV.  Differences  of  Trigonometrical  Lines        -  ib. 

V.  Resolution  of  Right-angled  Spherical  Triangles  ib. 

VI.  Resolution  of  Oblique-angled  Spherical  Triangles      -        -        345 


XVI  TABLE    OF    CONTENTS. 

PAOB 

INVESTIGATION  OP  ASTRONOMICAL  FORMULJE  -  348 

Formulae  for  the  Parallax  in  Right  Ascension  and  Declination, 

and  in  Longitude  and  Latitude  -  ib. 

Formulas  for  the  Aberration  in  Longitude  and  Latitude,  and  in 

Right  Ascension  and  Declination  -  355 

Formulae  for  the  Nutation  in  Right  Ascension  and  Declination  359 
Formulae  for  computing  the  effects  of  the  Oblateness  of  the 

Earth's  Surface,  upon  the  Apparent  Zenith  Distance  and 

Azimuth  of  a  Star      --------        353 

Solution  of  Kepler's  Problem,  by  which  a  Body's  Place  is  found 

in  an  Elliptical  Orbit 364 


NOTES. 

L  to  XXII -  369-382 


CALIFORNIA- 


AN 

ELEMENTARY    TREATISE 


ON 


ASTBOIOMY. 


INTRODUCTION. 

GENERAL  NOTIONS GENERAL  PHENOMENA  OF  THE  HEAVENS.^ 

1 .  The  space,  indefinite  in  extent,  which  is  exterior  to  the  earth, 
is  called  the  Heaven  or  Heavens,  or  the  Firmament.     The  sun, 
moon,  and  stars,  the  luminous  bodies  which  are  posited  in  this 
space,  are  called  the  Heavenly  Bodies.     The  entire  assemblage 
of  these  bodies  is  frequently  called  the  Heavens. 

2.  The  most  casual  observation  shows  us  that  the  heavenly 
bodies  are  subject  to  a  variety  of  motions,  as  well  as  to  various 
changes  of  appearance.     The  science  which  treats  of  the  laws 
and  causes  of  these  motions  and  changes,  is  called  Astronomy  ;* 
or,  more  particularly,  Astronomy  is  a  mixed  mathematical  science, 
which  treats  of  the  motions,  positions,  distances,  appearances,  mag- 
nitudes, and  physical  constitution  of  the  heavenly  bodies.     It  has 
been  divided  into  the  two  departments  of  Plane  or  Pure  Astronomy, 
a.nd  Physical  Astronomy.     Plane  Astronomy  comprehends,  1st,  the 
description    of  the    motions,   appearance^  and:  structure  of  the 
heavenly  bodies,  and  the  description  and  explanation  of  their  phe- 
nomena, which  may  be  called  Descriptive  Astronomy ;  2d,  the 
methods  of  observation  and  calculation  employed  in  obtaining  a 
knowledge  of  the  facts  embodied  in  Descriptive  Astronomy,  and 
the  computation  from  these  of  the  details  of  occasional  phenome- 
na, as  eclipses  of  the  sun  and  moon,  occupations  of  the- stars,.  &c., 
which  is  denominated  Practical  Astronomy <..  Physical  Astronomy 
investigates  inductively  the  physical  causes  of  the  observed  mo- 
tions and  constitution  of  the  great  bodies  of  the  material  universe, 
and  deduces,  as  a  mechanical  problem,  from  the  one  great  cause, 
the  principle  of  universal  gravitation,  all  the  minutiae  of  the  celes- 
tial mechanism. 

*  From  Affi^p,  a  star,  and  w.w,  a  law. 
1 


2  INTRODUCTION. 

3.  The  origin  of  the  science  of  Astronomy  is  involved  in  obscurity ;  but  it  it 
supposed  that  its  first  truths  were  discovered  in  the  early  ages  of  the  world  by 
shepherds,  who,  at  the  same  time,  watched  their  flocks  by  night,  and  followed  the 
motions  and  noted  the  varying  aspects  o'f  the  heavenly  bodies.     Each  successive 
age,  from  that  time  to  the  present,  has,  with  occasional  interruptions,  brought  to  it 
its  contributions  of  observations  and  discoveries.     The  imposing  character  of  the 
celestial  phenomena,  and  their  intimate  relations  to  the  every-day  wants  of  life,  as 
well  as  the  superstitions  of  the  ignorant,  have,  from  time  immemorial,  conspired 
to  attract  to  this  science  the  interested  attention  of  mankind,  and  promote  its  ad- 
vancement.    From  the  very  nature  of  things,  some  of  its  truths  have  only  unfolded 
themselves,  as  century  after  century  has  passed  away ;  while  others  still  await  the 
lapse  of  future  ages.    Its  history,  in  a  theoretical  point  of  view,  presents  two  prom- 
inent epochs,  viz  :  1.  The  epoch  of  the  discovery  of  the  true  system  of  the  world,  by 
Copernicus,  towards  the  middle  of  the  sixteenth   century ;  soon  followed  by  the 
discovery  of  the  exact  laws  of  its  motions  in  space,  by  Kepler,  (early  in  the  sev- 
enteenth century  ;)  which  has  so  completely  changed  the  whole  face  of  the  science, 
and  has  been  succeeded  by  such  a  mass  of  observations  of  greatly  increased  accu- 
racy, and   such  an   uninterrupted  series  of    important  discoveries,  that  it  may 
almost  be  said  to  be  the  date  of  its  origin,  as  the  science  is  now  taught.     2.  The 
epoch  of  the  discovery  of  universal  gravitation,  by  Sir  Isaac  Newton,  (1683;)  a 
discovery  that  has  brought  Astronomy  within  the  province  of  Mechanical  Philoso- 
phy, and  contributed  greatly  to  its  advancement  and  extension,  by  making  known 
its  physical  theory,  which  has  been  developed  by  Laplace  and  others   with  great 
minuteness  of  detail.     Contemplating  the  science  from  a  practical  point  of  view, 
we  find  that  its  most  prominent  epoch  is  that  of  the  discovery  of  the  telescope,  a 
the  beginning  of  the  seventeenth  century,  since  which  time,  by  the  adaptation  of 
the  telescope  to  instruments  for  admeasurement,   and  the  improvement  of  these 
instruments,  its  means  of  research  have  been  gradually  perfected  and  extended, 
as  art  and  science  have  advanced  hand  in  hand  :  until  from  a  few  shepherds,  un- 
der the  open  sky  on  the  plains  of  Chaldea,  with  naught  but  their  natural  powers 
of  vision,  there  has  come  to  be  a  large  body  of  professed  Astronomers  in  charge  of 
permanent  observatories  erected  in  almost  every  civilized  country  on  the  globe  ; 
and  furnished  at  the  same  time  with  telescopes  that  bring  the  heavenly  bodies  hun- 
dreds or  even  thousands  of  times  nearer,  and  disclose  a  new  world  of  celestial  ob- 
jects, and  with  instruments  that  mark  out,  with  the  greatest  precision,  the  ever 
varying  places  of  all  these  bodies. 

4.  To   be   able   to  form   correct  notions   of  the   phenomena 
of  the  heavens,  it  is  necessary  to  know  the  form  of  the  earth. 
We  learn   from  the  following  circumstances   that  the    earth   is 
a  body  of  a  globular  form,   insulated   in   space.     1st.  When  a 
vessel  is   receding  from  the  land,  an  observer    stationed   upon 

Fig.  1.  the    coast,  first  loses  sight 

of  the  hull,  then  of  the 
lower  parts  of  the  sails,  and 
lastly,  of  the  topsails.  This 
is  the  case  whatever  is  the 
direction  of  the  course  of 
the  vessel,  and  at  whatever 
part  of  the  earth  it  is  ob- 
served. That  this  is  a  proof 
of  the  roundness  of  the  sea, 
will  at  once  be  seen  on  in- 
specting Fig.  1.  It  will 
readily  be  perceived  that  no 
part  of  the  earth  could  be- 
come interposed  between  the 


GENERAL    NOTIONS. 


hull  and  the  lower  parts  of  the  sails  of  a  distant  vessel,  and  the  eye 
of  the  observer,  if  the  sea  were  really  what  it  appears  to  be,  an 
indefinitely  extended  plane.  2d.*  At  sea  the  visible  horizon,  or  the 
line  bounding  the  visible  portion  of  the  earth's  surface,  is  every- 
where a  circle,  of  a  greater  or  less  extent,  according  to  the  altitude 
of  the  point  of  observation,  and  is  on  all  sides  equally  depressed. 
To  illustrate  this  proof,  let  BOA  (Fig.  2)  represent  a  portion 


of  the  earth's  surface  supposed  to  be  spherical,  P  the  position  of 
the  eye  of  'the  observer,  and  DPC  a  horizontal  line.  If  we 
conceive  lines,  such  as  PA  and  PB,  to  be  drawn  through  the 
point  of  observation  P,  tangent  to  the  earth  in  every  direction,  it  is 
plain  that  these  lines  will  all  touch  the  earth  at  the  same  distance 
from  the  observer,  and  therefore  that  the  line  AGB,  conceived  to 
be  traced  through  all  the  points  of  contact,  A,B,  &c.,  which  would 
be  the  visible  horizon,  is  a  circle.  It  is  also  manifest  that  the  an- 
gles of  depression  CPA,  DPB,  &c.,  of  the  horizon  in  different 
directions,  would  be  equal,  and  that  a  greater  portion  of  the  earth's 
surface  would  be  seen,  and  thus  that  the  horizon  would  increase  in 
extent,  in  proportion  as  the  altitude  of  the  point  of  observation,  P, 
increased.  3d.  Navigators,  as  it  is  well  known,  have  sailed  around 
the  earth  in  different  directions.  These  facts  prove  the  surface  of 
the  sea  to  be  convex,  and  the  surface  of  the  land  conforms  very 
nearly  to  that  of  the  sea ;  for  the  elevations  of  the  highest  moun- 
tains bear  an  exceedingly  small  proportion  to  the  dimensions  of  the 
whole  earth. 

5.  If  an  indefinite  number  of  lines,  PA,  PB,  &c.,  be  conceived 
to  be  drawn  through  the  point  of  observation  P,  (Fig.  2,)  touching 
the  earth  on  all  sides,  a  conical  surface  will  be  formed,  having  its 
vertex  at  P,  and  extending  indefinitely  into  space.  All  heavenly 
bodies,  which  at  any  time  are  situated  below  this  surface,  have  the 
earth  interposed  between  them  and  the  eye  of  the  observer,  and 
therefore  cannot  be  seen.  All  bodies  that  are  above  this  surface, 
which  send  sufficient  light  to  the  eye,  are  visible.  That  portion 


4  INTRODUCTION. 

of  the  heavens  which  is  above  this  surface,  presents  the  appear- 
ance of  a  solid  vault  or  canopy,  resting  upon  the  earth  at  the  visi- 
ble horizon,  (see  Fig.  2 ;)  and  to  this  vault  the  sun,  moon,  and  stars 
seem  to  be  attached.  It  is  hardly  necessary  to  remark  that  this  is 
an  optical  illusion.  It  will  be  seen  in  the  sequel  that  the  heavenly 
bodies  are  distributed  through  space  at  various  distances  from  the 
earth,  and  that  the  distances  of  all  of  them  are  very  great  in  com- 
parison with  the  dimensions  of  the  earth. 

It  will  suffice,  in  the  conception  of  phenomena,  to  suppose  the 
eye  of  the  observer  to  be  near  the  earth's  surface,  and  that  the 
imaginary  conical  surface  above  mentioned,  which  separates  the 
visible  from  the  invisible  portion  of  the  heavens,  is  a  horizontal 
plane,  confounded  for  a  certain  distance  with  the  visible  part  of  the 
earth.  This  is  called  the  plane  of  the  horizon,  and  sometimes  the 
horizon  simply. 

6.  Up  and  downed!  any  place  on  the  earth's  surface,  are  from 
and  towards  the  swfece ;  and  thus  at  different  places  have  every 
variety  of  absolute  direction  in  space. 

This  fact  should  not  merely  be  acknowledged  to  be  true,  but  should  be  dwelt 
upon  until  the  mind  has  become  familiarized  to  the  conception  of  it,  and  divested, 
as  far  as  possible,  of  the  notion  of  an  absolute  up  and  down  in  space. 

7.  The  earth  is  surrounded  with  a  transparent  gaseous  medium, 
called  the  earth's  atmosphere,  estimated  to  be  some  fifty  miles  in 
height ;  through  which  all  the  heavenly  bodies  are  seen.     The  at- 
mosphere is  not  perfectly  transparent,  but  shines  throughout  with 
light  received  from  the  heavenly  bodies,  and  reflected  from  its  par- 
ticles ;  and  thus  forms  a  luminous  canopy  over  our  heads  by  day 
and  by  night.     This  is  called  the  sky.     It  appears  blue  because 
this  is  the  color  of  the  atmosphere ;  that  is,  because  the  particles 
of  the  atmosphere  reflect  the  blue  rays  more  abundantly  than  any 
other  color.     By  day  the  portion  of  the  atmosphere  which  lies 
above  the  horizon  is  highly  illuminated  by  the  sun,  and  shines  with 
so  strong  a  light  as  to  efface  the  stars. 

8.  The  most  conspicuous  of  the  celestial  phenomena,  is  a  con 
tinual  motion  common  to  all  the  heavenly  bodies,  by  which  they 
are  carried  around  the  earth  in  regular  succession.     The  daily 
circulation  of  the  sun  and  moon  about  the  earth  is  a  fact  recog- 
nised by  all  persons.     If  the  heavens  be  attentively  watched  on 
any  clear  evening,  it  will  soon  be  seen  that  the  stars  have  a  motion 
precisely  similar  to  that  of  the  sun  and  moon.     To  describe  the 
phenomenon  in  detail,  as  witnessed  at  night : — if,  on  a  clear  night, 
we  observe  the  heavens,  we  shall  find  that  the  stars,  while  they 
retain  the  same  situations  with  respect  to  each  other,  undergo  a 
continual  change  of  position  with  respect  to  the  earth.     Some  will 
be  seen  to  ascend  from  a  quarter  called  the  East,  being  replaced 
by  others  that  come  into  view,  or  rise ;  others,  to  descend  towards 
the  opposite  quarter,  the  West,  and  to  go  out  of  view,  or  set :  and 
if  our  observations  be  continued  throughout  the  night,  with  the 


GENERAL  PHENOMENA  OF  THE  HEAVENS.  5 

east  on  our  left,  and  the  west  on  our  right,  the  stars  which  rise  in 
the  east  will  be  seen  to  move  in  parallel  circles,  entirely  across  the 
visible  heavens,  and  finally  to  set  in  the  west.  Each  star  will 
ascend  in  the  heavens  during  the  first  half  of  its  course,  and  de- 
scend during  the  remaining  half.  The  greatest  heights  of  the 
several  stars  will  be  different,  but  they  will  all  be  attained  towards 
that  part  of  the  heavens  which  lies  directly  in  front,  called  the 
South.  If  we  now  turn  our  backs  to  the  south,  and  direct  our 
attention  to  the  opposite  quarter,  the  North,  new  phenomena  will 
present  themselves.  Some  stars  will  appear,  as  before,  ascending, 
reaching  their  greatest  heights,  and  descending ;  but  other  stars 
will  be  seen,  farther  to  tne  north,  that  never  set,  and  which  appear 
to  revolve  in  circles,  from  east  to  west,  about  a  certain  star  that 
seems  to  remain  stationary.  This  seemingly  stationary  star  is 
called  the  Pole  Star;  and  those  stars  that  revolve  about  it,  and  never 
set,  are  called  Circumpolar  Stars.  It  should  be  remarked,  how 
ever,  that  the  pole  star,  when  accurately  observed  by  means  of 
instruments,  is  found  not  to  be  strictly  stationary,  but  to  describe 
a  small  circle  about  a  point  at  a  little  distance  from  it  as  a  fixed 
centre.  This  point  is  called  the  North  Pole.  It  is,  in  reality, 
about  the  north  pole,  as  thus  defined,  and  not  the  pole  star,  that 
the  apparent  revolutions  of  the  stars  at  the  north  are  performed. 
At  the  corresponding  hours  of  the  following  night  the  aspect  of  the 
heavens  will  be  the  same,  from  which  it  appears  that  the  stars  re- 
turn to  the  same  position  once  in  about  24  hours.  It  would  seem, 
then,  that  the  stars  all  appear  to  move  from  east  to  west,  exactly 
as  if  attached  to  the  concave  surface  of  a  hollow  sphere,  which 
rotates  in  this  direction  about  an  axis  passing  through  the  station 
of  the  observer  and  the  north  pole  of  the  heavens,  in  a  space  of 
time  nearly  equal  to  24  hours.  For  the  sake  of  simplicity  this 
conception  is  generally  adopted.  This  motion,  common  to  all  the 
heavenly  bodies,  is  called  their  Diurnal  Motion.  It  is  ascertained, 
by  certain  accurate  methods  of  observation  and  computation,  here- 
after to  be  exhibited,  that  the  diurnal  motion  of  the  stars  is  strictly 
uniform  and  circular. 

9.  It  is  important  to  observe,  that  the  conception  of  a  single  sphere 
to  which  the  stars  are  supposed  to  be  attached,  will  not  represent 
their  diurnal  motion,  as  seen  from  every  part  of  the  earth's  surface^ 
unless  the  sphere  be  supposed  to  be  of  such  vast  dimensions  that 
the  earth  is  comparatively  but  a  mere  point  at  its  centre.* 

10.  A  circle  cut  out  of  the  heavens  conceived  to  be  a  rotating 
sphere,  by  a  plane  passing  through  the  axis  of  rotation,  has  a  north 

*  The  student  should  strive  to  familiarize  his  mind  with  this  notion  of  the  sphere 
of  the  heavens.  The  disposition,  so  natural  to  every  one,  to  conceive  the  stars  to 
be  at  no  very  great  distance  from  the  earth,  in  comparison  with  the  dimensions  of 
so  large  a  body,  will,  until  it  is  overcome,  often  give  rise  to  very  erroneous  concep- 
tions of  the  different  appearances  of  the  same  phenomenon,  as  viewed  from  different 
points  of  the  earth's  surface 


6  INTRODUCTION. 

and  south  direction  ;  and  a  circle  cut  out  by  a  plane  perpendicular 
to  the  axis,  has  an  east  and  west  direction. 

11.  The  greater  number  of  the  stars  preserve  constantly  the 
same  relative  position  with  respect  to  each  other  ;  and  they  are 
therefore  called  Fixed  Stars.  There  are,  however,  a  few  stars, 
called  Planets*  which  are  perpetually  changing  their  places  in  the 
heavens.  The  number  of  the  planets  is  ten.  Each  has  a  distinc- 
tive name,  as  follows  :  Mercury,  Venus,  Mars,  Jupiter,  Saturn, 
Uranus,  Ceres,  Pallas,  Juno,  and  Vesta.  Mercury,  Venus,  Mars, 
Jupiter,  and  Saturn  are  visible  to  the  naked  eye,  and  have  been 
known  from  the  most  ancient  times.  The  other  five,  namely,  Ura- 
nus, Ceres,  Pallas,  Juno,  and  Vesta,  cannot  be  seen  without  the 
assistance  of  the  telescope,  and  were  discovered  by  modern  ob- 
servers.! (See  Note  I,  at  the  end  of  the  Appendix.) 

1  2.  The  planets  are  distinguishable  from  each  other  either  by  a  dif- 
ference of  aspect,  or  by  a  difference  of  apparent  motion  with  respect 
to  the  sun.  Venus  and  Jupiter  are  the  two  most  brilliant  planets  : 
they  are  quite  similar  in  appearance,  but  their  apparent  motions 
with  respect  to  the  sun  are  very  different.  Venus  never  recedes 
beyond  40°  or  50°  from  the  sun,  while  Jupiter  is  seen  at  every  va- 
riety of  angular  distance  from  him.  Mars  is  known  by  the  ruddy 
color  of  his  light.  Saturn  has  a  pale,  dull  aspect. 

13.  The  apparent  motion  of  the  planets  is  generally  directed 
towards  the  east  ;  but  they  are  occasionally  seen  moving  towards 
the  west.     As  their  easterly  prevails  over  their  westerly  motion, 
they  all,  in  process  of  time,  accomplish  a  revolution  around  the 
earth.     The  periods  of  revolution  are  different  for  each  planet. 

14.  The  Sun  and  Moon  are  also  continually  changing  their 
places  among  the  fixed  stars. 

15.  From  repeated  examinations  of  the  situation  of  the  moon 
among  the  stars,  it  is  found  that  she  has  with  respect  to  them  a 
progressive  circular  motion  from  west  to  east,  and  completes  a  re- 
volution around  the  earth  in  about  27  days. 

16.  The  motion  of  the  sun  is  also  constantly  progressive,  and 
directed  from  west  to  east.     This  will  appear  on  observing  for  a 
number  of  successive  evenings  the  stars  which  first  become  visi- 
ble in  that  part  of  the  heavens  where  the  sun  sets.  It  will  be  found 
that  those  stars  which  in  the  first  instance  were  observed  to  set 
just  after  the  sun,  soon  cease  to  be  visible,  and  are  replaced  by 
others  that  were  seen  immediately  to  the  east  of  them  ;  and  that 
these,  in  their  turn,  give  place  to  others  situated  still  farther  to  the 
east.     The  sun,  then,  is  continually  approaching  the  stars  that  lie 


*  F»om  .wAov^rijf,  a  wanderer. 

t  The  planet  Uranus  was  discovered  in  1781  by  Dr.  Herschel,  who  gave  it  the 
name  of  Georgium  Sidus.  By  the  European  astronomers  it  was  called  Herschel. 
It  is  now  generally  known  by  the  name  given  in  the  text.  Ceres,  Pallas,  Juno, 
and  Vesta  have  been  discovered  since  1800  ;  the  first  by  Piazzi,  the  second  and 
fourth  by  Olbers,  and  the  third  by  Harding. 


GENERAL  PHENOMENA  OF  THE  HEAVENS. 


Fig.  3. 


on  the  eastern  side  of 
him.  To  make  this 
more  evident,  let  us 
suppose  that  the  small 
circle  aon  (Fig.  3)  rep- 
resents a  section  of 
the  earth  perpendic- 
ular to  the  axis  of  ro- 
tation of  the  imagi- 
nary sphere  of  the 
heavens,  (8,*)  con- 
ceived to  pass  through 
the  earth's  centre ;  the 
large  circle  H  Z  S  a 
section  of  the  heavens 
perpendicular  to  the 
same  line,  and  pass- 
ing through  the  sun ; 
and  the  right  line 
H  o  r  the  plane  of  the  horizon  at  the  station  o.  The  direction  of 
the  diurnal  motion  is  from  H  towards  Z  and  S.  Suppose  that  an 
hour  or  so  after  sunset  the  sun  is  at  S,  and  that  the  star  r  is  seen 
in  the  western  horizon ;  also  that  the  stars  t,  u,  v,  &c.,  are  so  dis- 
tributed that  the  distances  rt,  tu,  uv,  &c.  are  each  equal  to  S  r. 
Then,  at  the  end  of  two  or  three  weeks,  an  hour  after  sunset  the 
star  t  will  be  in  the  horizon ;  at  the  end  of  another  interval  of  two 
or  three  weeks  the  star  u  will  be  in  the  same  situation  at  the  same 
hour ;  at  the  end  of  another  interval,  the  star  v,  &c.  It  is  plain, 
then,  that  the  sun  must  at  the  ends  of  these  successive  intervals  be 
in  the  successive  positions  in  the  heavens,  r,  t,  u,  &c. ;  otherwise, 
when  he  is  brought  by  his  diurnal  motion  to  the  point  S,  below  the 
horizon,  the  stars  t,  u,  v,  &c.,  could  not  be  successively  in  the 
plane  of  the  horizon  at  r.  Whence  it  appears  that  he  has  a  mo- 
tion in  the  heavens  in  the  direction  S  r  t  u  v,  opposite  to  that  of 
the  diurnal  motion ;  that  is,  towards  the  east. 

Another  proof  of  the  progressive  motion  of  the  sun  among  the 
stars  from  west  to  east,  is  found  in  the  fact  that  the  same  stars  rise 
and  set  earlier  each  successive  night,  and  week,  and  month  during 
the  year.  At  the  end  of  six  months  the  same  stars  rise  and  set  12 
hours  earlier ;  which  shows  that  the  sun  accomplishes  half  a  revo- 
lution in  this  interval.  At  the  end  of  a  year,  or  of  365  days,  the 
stars  rise  and  set  again  at  the  same  hours,  from  which  it  appears 
that  the  sun  completes  an  entire  revolution  in  the  heavens  in  this 
period  of  time. 

It  is  to  be  observed  that  the  sun  does  not  advance  directly  to- 
wards the  east.  He  has  also  some  motion  from  south  to  north,  and 

*  Numbers  thus  enclosed  in  a  parenthesis  refer,  in  general,  to  a  previous  article. 


8  INTRODUCTION. 

north  to  south.  It  is  a  matter  of  common  observation  that  the  snn 
is  moving  towards  the  north  from  winter  to  summer,  and  towards 
the  south  from  summer  to  winter. 

17.  When  the  place  of  the  sun  in  the  heavens  is  accurately 
found  from  day  to  day  by  certain  methods  of  observation,  hereaf- 
ter to  be  explained,  it  appears  that  his  path  is  an  exact  circle,  in- 
clined about  23°  to  a  circle  running  due  east  and  west,  (10.) 

18.  The  motions  of  the  sun,  moon,  and  planets  are  for  the  most 
part  confined  to  a  certain  zone,  of  about  18°  in  breadth,  extending 
around  the  heavens  from  west  to  east,  (or  nearly  so,)  which  has 
received  the  name  of  the  Zodiac. 

19.  There  is  yet  another  class  of  bodies,  called  Comets*  (or 
hairy  Stars,)  that  have  a  motion  among  the  fixed  stars.    They  ap- 
pear only  occasionally  in  the  heavens,  and  continue  visible  only 
for  a  few  weeks  or  months.     They  shine  with  a  diffusive  nebu- 
lous light,  and  are  commonly  accompanied  by  a  fainter  divergent 
stream  of  similar  light,  called  a  tail. 

20.  The  motions  of  the  comets  are  not  restricted  to  the  zodiac. 
These  bodies  are  seen  in  all  parts  of  the  heavens,  and  moving  in 
every  variety  of  direction. 

21.  By  inspecting  the  planets  with  telescopes,  it  has  been  dis- 
covered that  some  of  them  are  constantly  attended  by  a  greater  or 
less  number  of  small  stars,  whose  positions  are  continually  vary- 
ing.    These  attendant  stars  are  called  Satellites.     The  planets 
which  have  satellites  are  Jupiter,  Saturn,  and  Uranus.     The  sat- 
ellites are  sometimes  called  Secondary  Planets ;  the  planets  upon 
which  they  attend  being  denominated  Primary  Planets. 

22.  The  sun  and  moon,  the  planets,  (including  the  earth,)  to- 
gether with  their  satellites,  and  the  comets,  compose  the  Solar 
System. 

23.  From  the  consideration  of  the  apparent  motions  and  other 
phenomena  of  the  solar  system,  several  theories  have  been  form- 
ed in  relation  to  the  arrangement  and  actual  motions  in  space  of 
the  bodies  that  compose  it.     The  theory,  or  system,  now  univer- 
sally received,  is  (in  its  most  prominent  features)  that  which  was 
taught  by  Copernicus  in  the  sixteenth  century,  and  which  is  known 
by  the  name  of  the  Copernican  System.     It  is  as  follows  : 

The  sun  occupies  a  fixed  centre,  about  which  the  planets  (in- 
cluding the  earth)  revolve  from  west  to  east,t  in  planes  that  are  but 
slightly  inclined  to  each  other,  and  in  the  following  order :  Mer- 
cury, Venus,  the  Earth,  Mars,  Vesta,  Juno,  Ceres,  Pallas,  Jupiter, 

*  From  Coma,  a  head  of  hair. 

t  A  motion  in  space  from  west  to  east  is  a  motion  from  right  to  left,  to  a  person 
situated  within  the  orbit  described,  and  on  the  north  side  of  its  plane.  To  obtain 
a  clear  conception  of  the  motions  of  the  solar  system,  the  reader  must  place  him- 
self, in  imagination,  in  some  such  situation  as  this,  entirely  detached  from  the 
earth  and  all  the  other  bodies  of  the  system.  It  is  customary  to  take  the  plane  of 
the  earth's  orbit  as  the  plane  of  reference  in  conceiving  of  the  planetary  motions. 


GENERAL  PHENOMENA  OF  THE  HEAVENS.  9 

Saturn,  and  Uranus.  The  earth  rotates  from  west  to  east,  about 
an  axis  inclined  to  the  plane  of  its  orbit  under  an  angle  of  about 
66J°,  and  which  remains  continually  parallel  to  itself  as  the  earth 
revolves  around  the  sun.  The  moon  revolves  from  west  to  east 
around  the  earth  as  a  centre ;  and,  in  like  manner,  the  satellites 
circulate  from  west  to  east  around  their  primaries.  Without  the 
solar  system,  and  at  immense  distances  from  it,  are  the  fixed  stars. 
(See  the  Frontispiece,  which  is  a  diagram  of  the  solar  'system  in 
projection.) 

24.  We  shall  here,  at  the  outset,  adopt  this  system  as  an  hypo- 
thesis, and  shall  rely  upon  the  simple  and  complete  explanations  it 
affords  of  the   celestial  phenomena,  as  they  come  to  be  investi- 
gated, together  with  the  evidence  furnished  by  Physical  Astrono- 
my, to  produce  entire  conviction  of  its  truth  in  the  mind  of  the 
student. 

25.  The  following  are  the  characters  or  symbols  employed  by 
astronomers  for  denoting  the  several  planets,  and  the  sun  and 
moon : — 

The  Sun, ©  Ceres,     .  .  .  .  ? 

Mercury, «  Pallas,     .  .  .  .  $ 

Venus,       5  Jupiter,  .  .  .  .  ^ 

The  Earth,     ....  0  Saturn,   .  .  .  .  ^ 

Mars,     ......  $  Uranus,  .  .  .  W 

Vesta, fi  The  Moon,  .  .  V 

Juno, $ 

26.  The  two  planets,  Mercury  and  Venus,  whose  orbits  lie  with- 
in the  earth's  orbit,  are  called  Inferior  Planets.     The  others  are 
called  Superior  Planets. 

27.  The  angular  distance  between  any  two  fixed  stars  is  found 
to  be  the  same,  from  whatever  point  on  the  earth's  surface  it  is 
measured.     It  follows,  therefore,  that  the  diameter  of -the  earth  is 
insensible,  when  compared  with  the  distance  of  the  fixed  stars ; 
and  that,  with  respect  to  the  region  of  space  which  separates  us 
from  these  bodies,  the  whole  earth  is  a  mere  point.    Moreover,  the 
angular  distance  between  any  two  fixed  stars  is  the  same  at  what- 
ever period  of  the  year  it  is  measured.     Whence,  if  the  earth  re- 
volves around  the  sun,  its  entire  orbit  must  be  insensible  in  com- 
parison with  the  distance  of  the  stars. 

28.  On  the  hypothesis  of  the  earth's  rotation,  the  diurnal  motion 
of  the  heavens  is  a  mere  illusion,  occasioned  by  the  rotation  of  the 
earth.    To  explain  this,  suppose  the  axis  of  the  earth  prolonged  on 
till  it  intersects  the  heavens,  considered  as  concentric  with  the 
earth.    Conceive  a  great  circle  to  be  tracgd  through  the  two  points 
of  intersection  and  the  point  directly  over  head,  and  let  the  position 
of  the  stars  be  referred  to  this  circle.     It  will  be  readily  perceived 
that  the  relative  motion  of  this  circle  and  the  stars  will  be  the  same, 
whether  the  circle  rotates  with  the  earth  from  west  to  east,  or,  the 

2 


10 


INTRODUCTION. 


earth  being  stationary,  the  whole  heavens  rotate  about  the  same 
axis  and  at  the  same  rate  in  the  opposite  direction.  Now,  as  the 
motion  of  the  earth  is  perfectly  equable,  we  are  insensible  of  it, 
and  therefore  attribute  the  changes  in  the  situations  of  the  stars 
with  respect  to  the  earth  to  an  actual  motion  of  these  bodies.  Ii 
follows,  then,  that  we  must  com  eive  the  heavens  to  rotate  as  above 
mentioned,  since,  as  we  have  seen,  such  a  motion  would  give  rise 
to  the  same  changes  of  situation  as  the  supposed  rotation  of  the 
earth.  It  was  stated  (Art.  8)  that  the  sphere  of  the  heavens  ap- 
pears to  rotate  about  a  line  passing  through  the  north  pole  and  the 
station  of  the  observer ;  but,  as  the  radius  of  the  earth  is  insensi- 
ble in  comparison  with  the  distance  of  the  stars,  an  axis  passing 
through  the  centre  of  the  earth  will,  in  appearance,  pass  through 
the  station  of  the  observer,  wherever  this  may  be  upon  the  earth's 
surface. 

29.  We  in  like  manner  infer  that  the  observed  motion  of  the 
sun  in  the  heavens  is  only  an  apparent  motion,  occasioned  by  the 
Fig.  4.  orbitual  motion  of  the  earth. 

Let  E,  E'  (Fig.  4)  represent 
two  positions  of  the  earth  in 
its  orbit  EE'E"  about  the 
sun  S.  When  the  earth  is 
at  E,  the  observer  will  refer 
the  sun  to  that  part  of  the 
heavens  marked  s;  but  when 
the  earth  is  arrived  at  E',  he 
will  refer  it  to  the  part  mark- 
ed s' ;  and  being  in  the  mean 
time  insensible  of  his  own 
motion,  the  sun  wall  appear 
to  him  to  have  described  in 
the  heavens  the  arc  s  s',  just 
the  same  as  if  it  had  actu- 
ally passed  over  the  arc  SS' 
in  space,  and  the  earth  had,  during  that  time,  remained  quiescent 
at  E.  The  motion  of  the  sun  from  5  towards  sf  will  be  from  west 
to  east,  since  the  motion  of  the  earth  from  E  towards  E'  is  in  this 
direction.  Moreover,  as  the  axis  of  the  earth  is  inclined  to  the 
plane  of  its  orbit  under  an  angle  of  66i°,  (23,)  the  plane  of  the 
sun's  apparent  path,  which  is  the  same  as  that  of  the  earth's  orbit, 
will  be  inclined  23|°  to  a  circle  perpendicular  to  the  earth's  axis, 
•r  to  a  circle  directed  due  east  and  west. 


PART    I. 

ON  THE   DETERMINATION   OF   THE   PLACES   AND  MOTIONS 
OF  THE  HEAVENLY  BODIES. 


CHAPTER    I. 

ON  THE  CELESTIAL  AND  TERRESTRIAL  SPHERES. 

30.  IN  determining  from  observation  the  apparent  positions  and 
motions  of  the  heavenly  bodies,  and,  in  general,  in  all  investigations 
that  have  relation  to  their  apparent  positions  and  motions,  Astron- 
omers conceive  all  these  bodies,  whatever  may  be  their  actual 
distance  from  the  earth,  to  be  referred  to  a  spherical  surface  of  an 
indefinitely  great  radius,  having  the  station  of  the   observer,  or 
what  comes  to  the  very  same  thing,  the  centre  of  the  earth,  for  its 
centre.     This  imaginary  spherical  surface  is  called  the  Sphere  of 
the  Heavens,  or  the  Celestial  Sphere.     It  is  important  to  observe, 
that  by  reason  of  the  great  dimensions  of  this  sphere,  if  two  lines 
be  drawn  through  any  two  points  of  the  earth,  and  parallel  to  each 
other,  they  will,  when  indefinitely  prolonged,  meet  it  sensibly  in 
the  same  point ;  and  that,  if  two  parallel  planes  be  passed  through 
any  two  points  of  the  earth,  they  will  intersect  it  sensibly  in  the 
same  great  circle.    This  amounts  to  saying  that  the  earth,  as  com- 
pared to  this  sphere,  is  to  be  considered  as  a  mere  point  at  its 
centre. 

31.  Not  only  is  the  size  of  the  earth  to  be  neglected  in  compari- 
son with  the  celestial  sphere,  but  also  the  size  of  the  earth's  orbit. 
Thus  the  supposed  annual  motion  of  the  earth  around  the   sun, 
does  not  change  the  point  in  which  a  line  conceived  to  pass  from 
any  station  upon  the  earth  in  any  fixed  direction  into  space,  pierces 
the  sphere  of  the  heavens  ;  nor  the  circle  in  which  a  plane  cuts  the 
same  sphere. 

The  fixed  stars  are  so  remote  from  the  earth  that  observers, 
wherever  situated  upon  the  earth,  and  in  the  different  positions  of 
the  earth  in  its  orbit,  refer  them  to  the  same  points  of  the  celestial 
sphere,  (27.)  The  other  heavenly  bodies  are  referred  by  observ- 
ers at  different  stations  to  points  somewhat  different. 

32.  For  the  purposes  of  observation  and  computation,  certain 
imaginary  points,  lines,  and  circles,  appertaining  to  the  celestial 
sphere,  are  employed,  which  we  shall  now  proceed  to  explain. 

(1.)  The  Vertical  Line,  at  any  place  on  the  earth's  surface,  is 


12 


ON    THE    CELESTIAL    SPHESE, 


the  line  of  descent  of  a  falling  body,  or  the  position  assumed  by  a 
plumb-line  when  the  plummet  is  freely  suspended  and  at  rest. 

Every  plane  that  passes  through  the  vertical  line  is  called  a  Ver- 
tical Plane.  Every  plane  that  is  perpendicular  to  the  vertical  line, 
is  called  a  Horizontal  Plane. 

(2.)  The  Sensible  Horizon  of  a  place  on  the  earth's  surface,  is 
the  circle  in  which  a  horizontal  plane,  drawn  through  the  place, 
cuts  the  celestial  sphere.  As  its  plane  is  tangent  to  the  earth,  it 
separates  the  visible  from  the  invisible  portion  of  the  heavens,  (5.) 
(3.)  The  Rational  Horizon  is  a  circle  parallel  to  the  former, 
the  plane  of  which  passes  through  the  centre  of  the  earth.  The 
zone  of  the  heavens  comprehended  between  the  sensible  and  ra- 
tional horizon  is  imperceptible,  or  the  two  circles  appear  as  one 
and  the  same  at  the  distance  of  the  earth,  (30.) 

(4.)  The  Zenith  of  a  place  is  the  point  in  which  the  vertical 
prolonged  upward  pierces  the  celestial  sphere.  The  point  in 
which  the  vertical,  when  produced  downward,  intersects  the  ce- 
lestial sphere,  is  called  the  Nadir. 

The  zenith  and  nadir  are  the  geometrical  poles  of  the  horizon. 
(5.)  The  Axis  of  the  Heavens  is  an  imaginary  right  line  pass- 
ing through  the  north  pole  (8)  and  the  centre  of  the  earth.  It  is 
the  line  about  which  the  apparent  rotation  of  the  heavens  is  per- 
formed. It  is,  also,  on  the  hypothesis  of  the  earth's  rotation,  the 
axis  of  rotation  of  the  earth  prolonged  on  to  the  heavens. 

(6.)  The  South  Pole  of  the  heavens  is  the  point  in  which  the 
axis  of  the  heavens  meets  the  southern  part  of  the  celestial  sphere. 

To  illustrate  the 
preceding  definitions, 
let  the  inner  circle 
n  O  s  (Fig.  5)  repre- 
sent the  earth,  and  the 
outer  circle  HZRN 
the  sphere  of  the 
heavens  ;  also  let  O 
be  a  point  on  the 
earth's  surface,  and 
OZ  the  vertical  line 
at  the  station  O. — 
Then  HOR  will  be 
the  plane  of  the  sen- 
sible horizon,  HCR 
the  plane  of  the  ra- 
tional horizon,  Z  the 
zenith,  and  N  the  na- 
dir ;  and  if  P  be  the 
north  pole  of  the  hea- 
vens, OP,  or  CP  its  parallel,  will  be  the  axis  of  the  heavens,  and 
P7  the  south  pole. 


DEFINITIONS 


13 


Fig.  6. 


The  lines  CP  and  OP  intersect  the  heavens  in  the  same  point, 
P;  and  the  planes  HOR,  and  HCR,  in  the  same  circle,  passing 
through  the  points  H  and  R. 

Unless  we  are  seeking  for  the  exact  apparent  place  in  the  heav- 
ens of  some  other  heavenly  body  than  a  fixed  star,  we  may  con- 
ceive the  observer  to  be  stationed  at  the  earth's  centre,  in  which 
case  OP  will  become  the  same 
as  CP,  and  HOR  the  same  as 
HCR  ;  as  represented  in  Fig.  6. 
In  this  diagram,  the  circle  of  the 
horizon  being  supposed  to  be  view- 
ed from  a  point  above  its  plane,  is 
represented  by  the  ellipse  HARez. 
Z  and  N  are  its  geometrical  poles. 
In  the  construction  of  Fig.  5  the 
eye  is  supposed  to  be  in  the  plane 
of  the  horizon,  and  HARa  is  pro- 
jected into  its  diameter  HCR. 

Every  different  place  on  the 
surface  of  the  earth  has  a  different 
zenith,  and,  except  in  the  case  of  diametrically  opposite  places,  a 
different  horizon.  To  illustrate  this,  let  nesq  (Fig.  7)  represent 
the  earth,  and  HZRP'  the  sphere  of  the  heavens  ;  then,  considering 
the  four  stations,  e,  O,  ft,  and  q,  the  zenith  and  horizon  of  the  first 


Fig.  7. 


will  be  respectively  E 
and  PeP' ;  of  the  se- 
cond Z  and  HOR  ;  of 
the  third  P  and  QnE  ; 
of  the  fourth  Q  and 
P'qP.  The  diametri- 
cally opposite  places 
0  and  O'  have  the 
same  rational  horizon, 
viz.  HCR.  The  same 
is  true  of  the  places  n 
and  s,  and  e  and  q. 
Their  rational  hori- 
zons are  respectively 
QCE  and  PCP'. 

(7.)  Vertical  Circles 
are  great  circles  pass- 
ing through  the  zenith 
and  nadir.  They  cut 
the  horizon  at  right  angles,  and  their  planes  are  vertical.  Thus, 
ZSM  (Fig.6)  represents  a  vertical  circle  passing  through  the  stai 
S,  called  the  Vertical  Circle  of  the  Star. 

(8.)  The  Meridian  of  a  place  is  that  vertical  circle  which  con- 


14  ON   THE    CELESTIAL    SPHERE. 

tains  the  north  and  south  poles  of  the  heavens.     The  plane  of  the 
meridian  is  called  the  Meridian  Plane. 

Thus,  PZRP'  is  the  meridian  of  the  station  C.  The  half 
HZR,  above  the  horizon,  is  termed  the  Superior  Meridian,  and 
the  other  half  RNH,  below  the  horizon,  is  termed  the  Inferior 
Meridian.  The  two  points,  as  H  and  R,  m  which  the  meridian 
cuts  the  horizon,  are  called  the  North  and  South  Points  of  the 
horizon ;  and  the  line  of  intersection,  as  HCR,  of  the  meridian 
plane  with  the  plane  of  the  horizon,  is  called  the  Meridian  Line, 
or'the  North  and  South  Line. 

(9.)  The  Prime  Vertical  is  the  vertical  circle  which  crosses  the 
meridian  at  right  angles.  It  cuts  the  horizon  in  two  points,  as 
e,  w,  called  the  East  and  West  Points  of  the  Horizon. 

(10.)  The  Altitude  of  any  heavenly  body  is  the  arc  of  a  vertical 
circle,  intercepted  between  the  centre  of  the  body  and  the  horizon, 
or  the  angle  at  the  centre  of  the  sphere,  measured  by  this  arc. 
Thus,  SM  or  MCS  is  the  altitude  of  the  star  S. 

(11.)  The  Zenith  Distance  of  a  heavenly  body  is  the  arc  of  a 
vertical  circle,  intercepted  between  its  centre  and  the  zenith ;  or 
the  distance  of  the  centre  of  the  body  from  the  zenith,  as  meas- 
ured by  the  arc  of  a  great  circle.  Thus,  ZS,  or  ZCS,  is  the 
zenith  distance  of  the  star  S. 

It  is  obvious  that  the  zenith  distance  and  altitude  of  a  body  are 
complements  of  each  other,  and  therefore  when  either  one  is  known 
that  the  other  may  be  found. 

(12.)  The  Azimuth  of  a  heavenly  body  is  the  arc  of  the  horizon, 
intercepted  between  the  meridian  and  the  vertical  circle  passing 
through  the  centre  of  the  body ;  or  the  angle  comprehended  be- 
tween the  meridian  plane  and  the  vertical  plane  containing  the 
centre  of  the  body.  It  is  reckoned  either  from  the  north  or  from 
the  south  point,  and  each  way  from  the  meridian.  HM  or  HCM 
represents  the  azimuth  of  the  star  S. 

The  Azimuth  and  Altitude,  or  azimuth  and  zenith  distance  of 
a  heavenly  body,  ascertain  its  position  with  respect  to  the  horizon 
and  meridian,  and  therefore  its  place  in  the  visible  hemisphere. 
Thus,  the  azimuth  HM  determines  the  position  of  the  vertical  cir- 
cle ZSM  of  the  star  S  with  respect  to  the  meridian  ZPH,  and  the 
altitude  MS,  or  the  zenith  distance  ZS,  the  position  of  the  star  in 
this  circle. 

(13.)  The  Amplitude  of  a  heavenly  body  at  its  rising,  is  the  arc 
of  the  horizon  intercepted  between  the  point  where  the  body  rises 
and  the  east  point.  Its.  amplitude  at  setting,  is  the  arc  of  the  ho- 
rizon intercepted  between  the  point  where  the  body  sets  and  the 
west  point.  It  is  reckoned  towards  the  north,  or  towards  the  south, 
according  as  the  point  of  rising  or  setting  is  north  or  south  of  the 
east  or  west  point.  Thus,  if  aBS A  represents  the  circle  described 
by  the  star  S  in  its  diurnal  motion,  ea  will  be  its  amplitude  al 
rising,  and  wA.  its  amplitude  at  setting. 


DEFINITIONS  15 

(14.)  The  Celestial  Equator,  or  the  Equinoctial,  is  a  great  cir- 
cle of  the  celestial  sphere,  the  plane  of  which  is  perpendicular  to 
the  axis  of  the  heavens.  The  north  and  south  poles  of  the  heav- 
ens are  therefore  its  geometrical  poles.  The  celestial  equator  is 
represented  in  Fig.  6  by  EwQe.  This  circle  is  also  frequently 
called  the  Equator,  simply. 

(15.)  Parallels  of  Declination  are  small  circles  parallel  to  the 
celestial  equator.  aBSA  represents  the  parallel  of  declination 
of  the  star  S. 

The  parallels  of  declination  passing  through  the  stars,  are  the 
circles  described  by  the  stars  in  their  apparent  diurnal  motion. 
These,  by  way  of  abbreviation,  we  shall  call  Diurnal  Circles. 

(16.)  Celestial  Meridians,  Hour  Circles,  and  Declination  Cir- 
cles, are  different  names  given  to  all  great  circles  which  pass 
through  the  poles  of  the  heavens,  cutting  the  equator  at  right  an- 
gles. PSP'  is  a  celestial  meridian.  The  angles  comprehended 
between  the  hour  circles  and  the  meridian,  reckoning  from  the 
meridian  towards  the  west,  are  called  Hour  Angles,  or  Horary 
Angles. 

(17.)  The  Ecliptic  is  that  great  circle  of  the  heavens  which  the 
sun  appears  to  describe  in  the  course  of  the  year. 

(18.)  The  Obliquity  of  the  Ecliptic  is  the  angle  under  which 
the  ecliptic  is  inclined  to  the  equator.  Its  amount  is  23  £°. 

(19.)  The  Equinoctial  Points  are  the  two  points  in  which  the 
ecliptic  intersects  the  equator.  That  one  of  these  points  which  the 
sun  passes  in  the  spring  is  called  the  Vernal  Equinox,  and  the 
other,  which  is  passed  in  the  autumn,  is  called  the  Autumnal  Equi- 
nox. These  terms  are  also  applied  to  the  epochs  when  the  sun  is 
at  the  one  or  the  other  of  these  points.  These  epochs  are,  for  the 
vernal  equinox  the  21st  of  March,  and  for  the  autumnal  equinox 
the  23d  of  September,  or  thereabouts. 

(20.)  The  Solstitial  Points  are  the  two  points  of  the  ecliptic 
90°  distant  from  the  vernal  and  autumnal  equinox.  The  one  that 
lies  to  the  north  of  the  equator  is  called  the  Summer  Solstice,  and 
the  other  the  Winter  Solstice.  The  epochs  of  the  sun's  arrival 
at  these  points  are  also  designated  by  the  same  terms.  The  sum- 
mer solstice  happens  about  the  21st  of  June,  and  the  winter  solstice 
about  the  22d  of  December. 

(21.)  The  Equinoctial  Colure  is  the  celestial  meridian  passing 
through  the  equinoctial  points  ;  and  the  Solstitial  Colure  is  the  ce- 
lestial meridian  passing  through  the  solstitial  points. 

(22.)  The  Polar  Circles  are  parallels  of  declination  at  a  distance 
from  the  poles  equal  to  the  obliquity  of  the  ecliptic.  The  one 
about  the  north  pole  is  called  the  Arctic  Circle ;  the  other,  about 
the  south  pole,  is  called  the  Antarctic  Circle. 

The  polar  circles  contain  the  geometrical  poles  of  the  ecliptic. 

(23.)  The  Tropics  are  parallels  of  declination  at  a  distance  from 
the  equator  equal  to  the  obliquity  of  the  ecliptic.  That  which  is 


16 


ON  THE  CELESTIAL  SPHERE. 


on  the  north  side  of  the  equator  is  called  the   Tropic  of  Cancer, 
and  the  other  the  Tropic  of  Capricorn. 

The  tropics  touch  the  ecliptic  at  the  solstitial  points. 

Fig.  8. 


Let  C  (Fig.  8)  represent  the  centre  of  the  earth  and  sphere, 
PCP'  the  axis  of  the  heavens,  EVQA  the  equator, -W VTA  the 
ecliptic,  and  K,  K',  its  poles.  Then  will  V  be  the  vernal  and  A 
the  autumnal  equinox  ;  W  the  winter,  and  T  the  summer  solstice ; 
P  VP'A  the  equinoctial  colure  ;  PKWK'T  the  solstitial  colure ; 
the  angle  TCQ,  or  its  measure  the  arc  TQ,  the  obliquity  of  the 
ecliptic;  KmU,  K'm'U',  the  polar  circles;  and  TrcZ,  Wrc'Z',  the 
tropics. 

It  is  important  to  observe  that,  agreeably  to  what  has  been  sta- 
ted, (Art.  30,)  the  directions  of  the  equator  and  ecliptic,  of  the  equi- 
noctial points,  and  of  the  other  points  and  circles  just  defined  and 
illustrated,  are  the  same  at  any  station  upon  the  surface  of  the 
earth  as  at  its  centre.  Thus,  the  equator  lies  always  in  the  plane 
passing  through  the  place  of  observation,  wherever  this  may  be, 
and  parallel  to  the  plane  which,  passing  through  the  earth's  centre, 
cuts  the  heavens  in  this  circle.  In  like  manner  the  ecliptic  lies, 
everywhere,  in  a  plane  parallel  to  that  which  is  conceived  to  pass 
through  the  centre  of  the  earth  and  cut  the  heavens  in  this  circle, 
and  so  for  the  other  circles. 

(24.)  The  Zodiac  (18)  extends  about  9°  on  each  side  of  the 
ecliptic. 


DEFINITIONS  17 

(25.)  The  ecliptic  and  zodiac  are  divided  into  twelve  equal  parts, 
called  Signs.  Each  sign  contains  30°.  The  division  commences 
at  the  vernal  equinox.  Setting  out  from  this  point,  and  following 
around  from  west  to  east,  the  Signs  of  the  Zodiac,  with  the  re- 
spective characters  by  which  they  are  designated,  are  as  follows  : 
Aries  T,  Taurus  8,  Gemini  n,  Cancer  Sj,  Leo  SI,  Virgo  W,  Li- 
bra =*«,  Scorpio  tn,  Sagittarius  /,  Capricornus  V3,  Aquarius  ss, 
Pisces  }£.  The  first  six  are  called  northern  signs,  being  north  of 
the  equinoctial.  The  others  are  called  southern  signs. 

The  vernal  equinox  corresponds  to  the  first  point  of  Aries,  and 
the  autumnal  equinox  to  the  first  point  of  Libra.  The  summer 
solstice  corresponds  to  the  first  point  of  Cancer,  and  the  winter 
solstice  to  the  first  point  of  Capricornus. 

The  mode  of  reckoning  arcs  on  the  ecliptic  is  by  signs,  degrees, 
minutes,  &c. 

A  motion  in  the  heavens  in  the  order  of  the  signs,  or  from  west 
to  east,  is  called  a  direct  motion,  and  a  motion  contrary  to  the  or- 
der of  the  signs,  or  from  east  to  west,  is  called  a  retrograde  mo- 
tion. 

(26.)  The  Right  Ascension  of  a  heavenly  body  is  the  arc  of  the 
equator  intercepted  between  the  vernal  equinox  and  the  declination 
circle  which  passes  through  the  centre  of  the  body,  as  reckoned 
from  the  vernal  equinox  towards  the  east.  It  measures  the  incli- 
nation of  the  declination  circle  of  the  body  to  the  equinoctial  col ure. 
Thus,  PSR  being  the  declination  circle  of  the  star  S,  and  V  the 
place  of  the  vernal  equinox,  VR  is  the  right  ascension  of  the  star. 
It  is  the  measure  of  the  angle  VPS.  If  PS'R'  be  the  declination 
circle  of  another  star  S',  SPS',  or  RR',  will  be  their  difference  of 
right  ascension. 

(27.)  The  Declination  of  a  heavenly  body  is  the  arc  of  a  circle 
of  declination,  intercepted  between  the  centre  of  the  body  and  the 
equator.  It  therefore  expresses  the  distance  of  the  body  from  the 
equator.  Thus,  RS  is  the  declination  of  the  star  S.. 

Declination  is  North  or  South,  according  as-  the  body  is  north  or 
south  of  the  equator. 

In  the  use  of  formulae,  a  south  declination  is  regarded  as  nega- 
tive. 

The  right  ascension  and  declination  of  a  heavenly  body  are  two 
co-ordinates,  which,  taken  together,  fix  its  position  in  the  sphere 
of  the  heavens :  for  they  make  known  its  situation  with  respect  to 
two  circles,  the  equinoctial  colure,  and  the  equator.  Thus,  VR 
and  RS  ascertain  the  position  of  the  star  S  with  respect  to  the  cir- 
cles PVP'A,  and  VQAE. 

(28.)  The  Polar  Distance  of  a  heavenly  body  is  the  arc  of  a  de- 
clination circle,  intercepted  between  the  centre  of  the  body  and  the 
elevated  pole.  The  polar  distance  is  the  complement  of  the  decli- 
nation, and,  therefore,  when  either  is  known  the  other  may  be 
found. 

3 


18  ON  THE  TERRESTRIAL  SPHERE. 

(29.)  Circles  of  Latitude  are  great  circles  of  the  celestial  sphere, 
which  pass  through  the  poles  of  the  ecliptic,  and  therefore  cut  this 
circle  at  right  angles.  Thus,  KSL  represents  a  part  of  the  circle 
of  latitude  of  the  star  S. 

(30.)  The  Longitude  of  a  heavenly  body  is  the  arc  of  the  eclip- 
tic, intercepted  between  the  vernal  equinox  and  the  circle  of  lati- 
tude which  passes  through  the  centre  of  the  body,  as  reckoned 
from  the  vernal  equinox  towards  the  east,  or  in  the  order  of  the 
signs.  It  measures  the  inclination  of  the  circle  of  latitude  of  the 
body  to  the  circle  of  latitude  passing  through  the  vernal  equinox. 
Thus,  VL  is  the  longitude  of  the  star  S.  It  is  the  measure  of  the 
angle  VKS. 

(31.)  The  Latitude  of  a  heavenly  body  is  the  arc  of  a  Circle  of 
latitude,  intercepted  between  the  centre  of  the  body  and  the  eclip- 
tic. It  therefore  expresses  the  distance  of  the  body  from  the  eclip- 
tic. Thus,  LS  is  the  latitude  of  the  star  S. 

Latitude  is  North  or  South,  according  as  the  body  is  north  or 
south  of  the  ecliptic. 

In  the  use  of  formulae,  a  south  latitude  is  affected  with  the  mi- 
nus sign. 

The  longitude  and  latitude  of  a  heavenly  body  are  another  set 
of  co-ordinates,  which  serve  to  fix  its  position  in  the  heavens.  They 
ascertain  its  situation  with  respect  to  the  circle  of  latitude  passing 
through  the  vernal  equinox  and  the  ecliptic.  Thus,  VL  and  LS 
fix  the  position  of  the  star  S,  making  known  its  situation  with  re- 
spect to  the  circles  KVK'A  and  VTAW. 

(32.)  The  Angle  of  Position  of  a  star,  is  the  angle  included  at 
the  star  between  the  circles  of  latitude  and  declination  passing 
through  it.  PSK  is  the  angle  of  position  of  the  star  S. 

(33.)  The  Astronomical  Latitude,  or  the  Latitude,  of  a  place,  is 
the  arc  of  the  meridian  intercepted  between  the  zenith  and  the  ce- 
lestial equator.  It  is  North  or  South,  according  as  the  zenith  is 
north  or  south  of  the  equator.  ZE  (Fig.  7)  represents  the  latitude 
of  the  station  O  ;  QOE  or  QCE  being  the  equator. 

33.  The  earth's  surface,  considered  as  spherical,  (which  ac- 
curate admeasurement,  upon  principles  that  will  be  explained  in 
the  sequel,  proves  it  to  be,  very  nearly,)  is  called  the  Terrestrial 
Sphere.  The  following  geometrical  constructions  appertain  to  the 
terrestrial  sphere,  as  it  is  employed  for  the  purposes  of  astronomy. 
It  will  be  observed  that  they  correspond  to  those  of  the  celestial 
sphere  above  described,  and  are  used  for  similar  objects. 

(1.)  The  North  and  South  Poles  of  the  Earth  are  the  two  points 
in  which  the  axis  of  the  heavens  intersects  the  terrestrial  sphere. 
They  are  also  the  extremities  of  the  earth's  axis  of  rotation. 

(2.)  The  Terrestrial  Equator  is  the  great  circle  in  which  a 
plane  passing  through  the  centre  of  the  earth,  and  perpendicular  to 
the  axis  of  the  heavens  and  earth,  cuts  the  terrestrial  sphere.  The 
terrestrial  and  the  celestial  equator,  then,  lie  in  the  same  plane. 


DEFINITIONS 


19 


The  poles  of  the  earth  are  the  geometrical  poles  of  the  terrestrial 
equator.  The  two  hemispheres  into  which  the  terrestrial  equator 
divides  the  earth,  are  called,  respectively,  the  Northern  Hemi- 
sphere and  the  Southern  Hemisphere. 

(3.)  Terrestrial  Meridians  are  great  circles  of  the  terrestrial 
sphere,  passing  through  the  north  and  south  poles  of  the  earth,  and 
cutting  the  equator  at  right  angles.  Every  plane  that  passes  through 
the  axis  of  the  heavens,  cuts  the  celestial  sphere  in  a  celestial  me- 
ridian^ and  the  terrestrial  sphere  in  a  terrestrial  meridian. 

Let  PP'  (Fig.  9)  represent  the  axis  of  the  heavens,  O  the  centre 
of  the  earth,  and  p  andp'  its  poles.  Then,  elq  will  represent  the 

Fig.  9. 


terrestrial  equator,  (ELQ  representing  the  celestial  equator;)  and 
pep'  andpsp'  terrestrial  meridians,  (PEP'  and  PSP'  representing 
celestial  meridians.) 

(4.)  The  Reduced  Latitude  of  a  place  on  the  earth's  surface  is 
the  arc  of  the  terrestrial  meridian,  intercepted  between  the  place 
and  the  equator,  or  the  angle  at  the  centre  of  the  earth  measured 
by  this  arc.  Thus,  oe,  or  the  angle  oOe,  is  the  reduced  latitude 
of  the  place  o.  Latitude  is  North  or  South,  according  as  the 
place  is  north  or  south  of  the  equator.  The  reduced  latitude  dif- 
fers somewhat  from  the  astronomical  latitude,  by  reason  of  the 
slight  deviation  of  the  earth  from  a  spherical  form.  Their  differ- 
ence is  called  the  Reduction  of  Latitude. 

(5.)  Parallels  of  Latitude  are  small  circles  of  the  terrestrial 


20  ON    THE    TERRESTRIAL    SPHERE. 

sphere  parallel  to  the  equator.   Every  point  of  a  parallel  of  latitude 
has  the  same  latitude. 

The  parallels  of  latitude  which  correspond  in  situation  with  the 
polar  circles  and  tropics  in  the  heavens,  have  received  the  same 
appellations  as  these  circles.  (See  defs.  22,  23,  p.  15.) 

(6.)  The  Longitude  of  a  place  on  the  earth's  surface,  is  the  in- 
clination of  its  meridian  to  that  of  some  particular  station,  fixed 
upon  as  a  circle  to  reckon  from,  and  called  the  First  Meridian.  It 
is  measured  by  the  arc  of  the  equator,  intercepted  between  the  first 
meridian  and  the  meridian  passing  through  the  place,  and  is  called 
East  or  West,  according  as  the  latter  meridian  is  to  the  east  or  to 
the  west  of  the  first  meridian.  Thus,  if  pqp'  be  supposed  to  re- 
present the  first  meridian,  the  angle  spq,  or  the  arc  ql,  will  be  the 
longitude  of  the  place  s. 

Different  nations  have,  for  the  most  part,  adopted  different  first 
meridians.  The  English  use  the  meridian  which  passes  through 
the  Royal  Observatory  at  Greenwich,  near  London;  and  the 
French,  the  meridian  of  the  Royal  Observatory  at  Paris.  In  the 
United  States  the  longitude  is,  for  astronomical  purposes,  reckoned 
from  the  meridian  of  Greenwich  or  Paris,  (generally  the  former.) 

The  longitude  and  latitude  of  a  place  designate  its  situation  on 
the  earth's  surface.  They  are  precisely  analogous  to  the  right  as- 
cension and  decimation  of  a  star  in  the  heavens. 

34.  The  diagram  (see  Fig.  6)  which  we  made  use  of  in  Art.  32 
in  illustrating  our  description  of  the  circles  of  the  celestial  sphere, 
represents  the  aspect  of  this  sphere  at  a  place  at  which  the  north 

pole  of  the  heavens  is  some- 
where between  the  zenith  arid 
horizon.  Such  is  the  position 
of  the  north  pole  at  all  places 
situated  between  the  equator 
and  the  north  pole  of  the 
earth.  For,  let  O  (Fig.  10) 
represent  a  place  on  the  earth's 
surface,  HOR  the  horizon, 
OZ  the  vertical,  HZR  the 
meridian, and  ZE  the  latitude. 
QOE  will  then  represent  the 
equinoctial,  and  P,  P',  90°  dis- 
tant from  E  and  on  the  meri- 
dian, the  poles.  Now,  we  have 
HP  =  ZH  —  ZP  =  90°  —  ZP  ;  ZE  =  PE  —  ZP  -  90°  —  ZP. 

Whence  HP  =  ZE. 

Thus,  the  altitude  of  the  pole  is  everywhere  equal  to  the  latitude 
of  the  place.  It  follows,  therefore,  that  in  proceeding  from  the 
equator  to  the  north  pole,  the  altitude  of  the  north  pole  of  the  heav- 
ens will  gradually  increase  from  0°  to  90°. 

By  inspecting  Fig.  7,  it  will  be  seen  that  this  increase  of  the  al- 


ASPECTS  OF  THE  CELESTIAL  SPHERE. 


21 


titude  of  the  pole  in  going  north,  is  owing  to  the  fact  that  in  fol- 
lowing the  curved  surface  of  the  earth  the  horizon,  which  is  con- 
tinually tangent  to  the  earth,  is  being  constantly  more  and  more 
depressed  towards  the  north,  while  the  absolute  direction  of  the 
pole  remains  unaltered. 

If  the  spectator  is  in  the  southern  hemisphere,  the  elevated  pole, 
as  it  is  always  on  the  opposite  side  of  the  zenith  from  the  equator, 
will  be  the  south  pole.  At  corresponding  situations  of  the  spec- 
tator it  will  obviously  have  the  same  altitude  as  the  north  pole  in 
the  northern  hemisphere. 

35.  Let  us  now  inquire  minutely  into  the  principal  circumstan- 
ces of  the  diurnal  motion  of  the  stars,  as  it  is  seen  by  a  spectator 
situated  somewhere  between  the  equator  and  the  north  pole.  And 
in  the  first  place,  it  is  a  simple  corollary  from  the  proposition  just 
established,  that  the  parallel  of  declination  to  the  north,  whose 
polar  distance  is  equal  to  the  latitude  of  the  place,  will  lie  entirely 
above  the  horizon,  and  just  touch  it  at  the  north  point.  This  cir- 
cle is  called  the  circle  of  perpetu-  Fig.  11. 
al  apparition ;  the  line #H  (Fig.  11) 
represents  its  projection  on  the  me- 
ridian plane.  The  stars  compre- 
hended between  it  and  the  north 
pole  will  never  set.  As  the  de- 
pression of  the  south  pole  is  equal 
to  the  altitude  of  the  north  pole,  H[ 
the  parallel  of  declination  o  R,  at 
,1  distance  from  the  south  pole 
equal  to  the  latitude  of  the  place, 
will  lie  entirely  below  the  horizon, 
and  just  touch  it  at  the  south  point. 
The  parallel  thus  situated  is  call- 
ed the  circle  of  perpetual  occultation.  The  stars  comprehended 
between  it  and  the  south  pole  will  never  rise. 

The  celestial  equator  (which  passes  through  the  east  and  west 
points)  will  intersect  the  meridian  at  a  point  E,  whose  zenith  dis- 
tance ZE  is  equal  to  the  latitude  of  the  place  (Def.  33,  Art.  32,)  and 
consequently,  whose  altitude  RE  is  equal  to  the  co-latitude  of  the 
place.  Therefore,  in  the  situation  of  the  observer  above  supposed, 
the  equator  QOE,  passing  to  the  south  of  the  zenith,  will,  togeth- 
er with  the  diurnal  circles  nr,  st,  &c.,  which  are  all  parallel  to  it, 
be  obliquely  inclined  to  the  horizon,  making  with  it  an  angle  equal 
to  the  co-latitude  of  the  place.  As  the  centres  c,c',  &c.,  of  the 
diurnal  circles  lie  on  the  axis  of  the  heavens,  which  is  inclined  to 
the  horizon,  all  diurnal  circles  situated  between  the  two  circles  of 
perpetual  apparition  and  occultation,  aH  and  oR,  with  the  excep- 
tion of  the  equator,  will  be  divided  unequally  by  the  horizon.  The 
greater  parts  of  the  circles  nr,  nY,  &c.,  to  the  north  of  the  equa- 
tor, will  be  above  the  horizon ;  and  the  greater  parts  of  the  circles 


22  ON  THE  CELESTIAL  SPHERE. 

sty  s't',  &c.,  to  the  south  of  the  equator,  will  be  below  the  horizon 
Therefore,  while  the  stars  situated  in  the  equator  will  remain  an 
equal  length  of  time  above  and  below  the  horizon,  those  to  the 
north  of  the  equator  will  remain  a  longer  time  above  the  horizon 
than  below  it ;  and  those  to  the  south  of  the  equator,  on  the  con- 
trary, a  longer  time  below  the  horizon  than  above  it.  It  is  also 
obvious,  from  the  manner  in  which  the  horizon  cuts  the  different 
diurnal  circles,  that  the  disparity  between  the  intervals  of  time  that 
a  star  remains  above  and  below  the  horizon,  will  be  the  greater  the 
more  distant  it  is  from  the  equator.  Again,  the  stars  will  all  cul- 
minate, or  attain  to  their  greatest  altitude,  in  the  meridian :  for, 
since  the  meridian  crosses  the  diurnal  circles  at  right  angles,  they 
will  have  the  least  zenith  distance  when  in  this  circle.  Moreover, 
as  the  meridian  bisects  the  portions  of  the  diurnal  circles  which  lie 
above  the  horizon,  the  stars  will  all  employ  the  same  length  of 
time  in  passing  from  the  eastern  horizon  to  the  meridian,  as  in 
passing  from  the  meridian  to  the  western  horizon.  The  circum- 
polar  stars  will  pass  the  meridian  twice  in  24  hours  ;  once  above, 
and  once  below  the  pole.  These  meridian  passages  are  called, 
respectively,  Upper  and  Lower  Culminations,  or  Inferior  and  Su- 
perior Transits. 

It  will  be  observed,  that  in  travelling  towards  the  north  the  cir- 
cles of  perpetual  apparition  and  occultation,  together  with  those 
portions  of  the  heavens  about  the  poles  which  are  constantly  visible 
and  invisible,  are  continually  on  the  increase. 

It  is  evident,  from  what  is  stated  in  Art.  34,  that  the  circum- 
stances of  the  diurnal  motion  will  be  the  same  at  any  place  in  the 
southern  hemisphere,  as  at  the  place  which  has  the  same  latitude 
in  the  northern. 

The  celestial  sphere  in  the  position  relative  to  the  horizon  which 
we  have  now  been  considering,  which  obtains  at  all  places  situated 
between  the  equator  and  either  pole,  is  called  an  Oblique  Sphere, 
because  all  bodies  rise  and  set  obliquely  to  the  horizon. 

Fig.  12.  36.  When  the  spectator  is  sit- 

uated on  the  equator,  both  the 
celestial  poles  will  be  in  his  hori- 
zon, (34,)  and  therefore  the  celes- 
tial equator  and  the  diurnal  circles 
in  general  will  be  perpendicular  to 
the  horizon.  This  situation  of  the 
sphere  is  called  a  Right  Sphere, 
for  the  reason  that  all  bodies  rise 
and  set  at  right  angles  with  the 
horizor .  It  is  represented  in  Fig. 
12.  As  the  diurnal  circles  are 
bisected  by  the  horizon,  the  stars 
will  all  remain  the  same  length  of 
time  above  as  below  the  horizon. 


ASTRONOMICAL  INSTRUMENTS. 


23 


37.  If  the  observer  be  at  either 
oi  the  poles,  the  elevated  pole  of 
the  heavens  will  be  in  his  zenith, 
(34,)  and  consequently,  the  celes- 
tial equator  will  be  in  his  horizon. 
The  stars  will  move  in  circles 
parallel  to  the  horizon,  and  the 
whole  hemisphere,  on  the  side  of 
the  elevated  pole,  will  be  continu- 
ally visible,  while  the  other  hem- 
isphere will  be  continually  invis- 
ible. This  is  called  a  Parallel 
Sphere.  It  is  represented 
Fig.  13. 


m 


Fig.  13. 


CHAPTER   II. 

ON  THE  CONSTRUCTION  AND  USE  OF  THE  PRINCIPAL  ASTRONOMICAL 

INSTRUMENTS. 

38.  ASTRONOMICAL  INSTRUMENTS  are,  for  the  most  part,  used  for 
the  admeasurement  of  arcs  of  the  celestial  sphere,  or  of  angles  cor- 
responding to  such  arcs  at  the  earth's  surface.  They  consist,  es- 
sentially, of  a  refracting  telescope  turning  upon  a  horizontal  axis, 
and  of  a  vertical  graduated  limb,  (or,  in  some  cases,  of  both  a  ver- 
tical and  a  horizontal  graduated  limb,)  to  indicate  the  angle  passed 
over  by  the  telescope.  At  the  common  focus  of  the  object-glass 
and  eye-glass  of  the  telescope  is  a  diaphragm,  or  circular  plate,  at- 
tached to  which  are  two  very  fine  wires,  or  spider-lines,  crossing 
each  other  at  right  angles  in  its  centre.  The  place  of  this  dia- 
phragm may  be  altered  by  adjusting  screws  ;  it  is  by  this  means 
brought  into  such  a  position  that  the  cross  of  the  wires  will  lie  on 
the  axis  of  the  telescope,  (that  is,  the  line  joining  the  centres  of  the 
object-glass  and  eye-glass.)  The  line  joining  the  centre  of  the  ob- 
ject-glass and  the  cross  of  the  wires,  is  technically  termed  the  Line 
of  Collimation.  Bringing  the  cross  of  the  wires  upon  the  axis  of 
the  telescope,  is  called  Adjusting  the  Line  of  Collimation.  A  star 
is  known  to  be  on  the  line  of  Collimation  when  it  is  bisected  by  the 
cross-wires. 

The  telescope  either  turns  around  the  centre  of  the  graduated 
limb,  or,  which  is  more  common,  the  limb  and  telescope  are  firmly 
attached  to  each  other,  and  turn  together.  In  the  first  arrange- 
ment a  small  steel  plate,  firmly  connected  with  the  telescope,  slides 
along  the  limb,  upon  this  plate  a  small  mark  is  drawn,  which  is 
called  the  Index.  The  required  angle  is  read  off  by  noting  the 


24 


ASTRONOMICAL  INSTRUMENTS. 


angle  upon  the  limb  which  is  pointed  out  by  the  index ;  the  zero 
on  the  limb  being  generally,  in  practice,  the  point  from  which  the 
angle  is  reckoi.ed.  When  the  telescope  and  graduated  limb  are 
firmly  connected,  the  limb  slides  past  the  index,  which  is  now  sta- 
tionary. The  limbs  of  even  the  largest  instruments  are  not  divided 
into  smaller  parts  than  about  5',  but,  by  means  of  certain  subsidi- 
ary contrivances,  the  angle  may,  with  some  instruments,  be  read 
off  lo  within  a  fraction  of  a  second. 

39.  The  principal  contrivances  for  increasing  the  accuracy  of 
the  reading  off  of  angles,  are  the  Vernier,  the  Micrometer  Screw,  and 
the  Micrometer  Microscope  or  Reading  Microscope.  The  Vernier 
is  only  the  index  plate,  so  graduated  that  a  certain  number  of  its 
divisions  occupy  the  same  space  as  a  number  one  less  on  the  limb. 
Fig.  14  represents  a  vernier  and  a  portion  of  the  limb  of  the  instru- 
ment, 15  divisions  on  the  vernier  corresponding  to  14  on  the  limb. 
If  we  suppose  the  smallest  divisions  of  the  limb  to  be  15',  and  call 
x  the  number  of  minutes  in  one  division  of  the  vernier,  then, 

15  x  ==  14  X  15',  and  x  =  14'. 

Thus,  the  difference  between  a  division  on  the  vernier  and  one 
on  the  limb,  will  be  1'.  Accordingly,  if  the  index,  which  is  the 
first. mark  on  the  vernier,  should  be  little  past  the  mark  40°  on  the 
limb,  and  the  second  mark  of  the  vernier  should  coincide  with  the 
next  point  of  division,  marked  15',  the  angle  would  be  40°  1'.  If 
the  third  mark  on  the  vernier  were  coincident  with  the  next  division 
of  the  limb,  marked  30',  the  angle  would  be  40°  2'.  If  the  fourth 
with  the  next  division  to  this,  40°  3' ;  and  so  on. 

By  making  the  divisions  on  the  vernier  more  numerous,  the  an- 
Fig.  14.  gle  can  be  read  off  with  greater 

precision ;  but  a  better  expedi- 
ent is  provided  in  the  Microme- 
ter Screw.  This  piece  of  me- 
chanism is  represented  in  Fig. 
14.  The  part  E  can  be  fast- 
ened to  the  limb  of  the  instru- 
ment by  means  of  a  screw.  FG 
is  a  screw,  with  a  milled  head  at 
F,  working  in  a  collar  fixed  in 
the  under  part  of  E ,  and  in  a  nut 
fixed  in  the  under  part  of  the  tel- 
escope T  t.  When  the  part  E 
is  fixed  or  clamped,  and  the 
screw  is  turned  around  by  its 
milled  head  at  F,  it  must  com- 
municate a  direct  motion  to  the 
nut,  and,  consequently,  to  the 

telescope  and  vernier  in  the  direction  of  FG.  Attached  to  the  screw, 
or  to  the  small  cylinder  on  which  it  is  formed,  is  an  index  D,  move- 
able  together  with  the  screw,  and  on  a  thin  graduated  immoveable 


READING  MICROSCOPE TIME,  ETC.  25 

plate,  the  profile  only  of  which  is  shown  in  the  figure.  Suppose 
now  that  the  screw  is  of  such  fineness  that  while,  together  with  the 
index  D,  it  makes  a  complete  revolution,  the  vernier  moves  through 
an  arc  of  1'.  Then,  if  the  plate  be  divided  into  60  equal  parts,  a 
motion  of  the  index  over  one  of  these  parts  would  answer  to  a  mo- 
tion of  I"  on  the  limb.  This  being  understood,  to  show  the  use  of 
the  micrometer  screw,  suppose  that  no  two  marks  on  the  vernier 
and  limb  are  coincident :  bring  the  two  nearest  into  coincidence  by 
turning  the  screw,  and  the  number  of  divisions  passed  over  by  the 
index  D  will  be  the  seconds  to  be  added  to  or  subtracted  from  the 
angle  read  off  with  the  vernier.  In  observing  the  coincidence  of 
the  divisions  of  the  limb  and  vernier,  the  eye  is  assisted  by  a  mi- 
croscope.* 

40.  The  Reading  Microscope  is  a  compound  microscope  firmly 
fixed  opposite  to  the  limb,  and  furnished  with  cross-wires  in  the  focus 
of  the  eye-glass,  or  conjugate  focus  of  the  object-glass,  moveable  by 
a  fine-threaded  micrometer  screw,  that  is,  a  screw  (such  as  was  de- 
scribed in  the  previous  article)  provided  with  an  immoveable  grad- 
uated circular  plate,  and  an  index  turning  with  the  screw,  and  glid- 
ing over  the  plate,  to  measure  the  exact  distance  through  which  the 
head  of  the  screw  is  moved.     The  observer  looks  through  the 
microscope  at  the  limb.  The  centre  of  the  microscope  corresponds 
to  the  index  of  a  fixed  vernier  plate.     By  turning  the  screw  the 
intersection  of  the  wires  is  moved  over  the  space  which  separates 
it  from  the  nearest  line  of  division  on  the  limb,  in  the  direction  of 
the  zero,  and  the  number  of  turns  and  parts  of  a  turn  of  the  screw 
being  noted  by  means  of  the  graduated  plate,  the  number  of  mi- 
nutes and  seconds  in  this  space  becomes  known.     The  minutes 
and  seconds  thus  found  being  added  to  the  angle  read  off  from  the 
limb,  the  result  will  be  the  angle  sought. 

41.  It  is  obvious  that,  other  things  being  the  same,  instruments 
are  accurate  in  proportion  to  the  power  of  the  telescope  and  the 
size  of  the  limb.     The  large  instruments  now  in  use  in  astronomi- 
cal observatories,  are  relied  upon  as  furnishing  angles  to  within  1" 
of  the  truth. 

42.  Time  is  an  essential  element  in  astronomical  observation. 
Three  different  kinds  of  time  are  employed  by  astronomers :  Si- 
dereal, Apparent  or  True  Solar,'  and  Mean  Solar  Time. 

43.  Sidereal  Time  is  time  as  measured  by  the  diurnal  motion 
of  the  stars,  or,  more  properly,  of  the  vernal  equinox.    A  Sidereal 
Day  is  the  interval  between  two  successive  meridian  transits  of  a 
star,  or,  (as  it  is  now  most  generally  considered,)  the  interval  be- 
tween two  successive  transits  of  the  vernal  equinox..  It  commences 
at  the  instant  when  the  vernal  equinox  is  on  the  superior  meridian, 
and  is  divided  into  24  Sidereal  Hours. 

44.  Apparent,  or  True  Solar  Time,  is  deduced  from  observa 

*  Weodhouse's  Astronomy,  v»l.  i.  p.  55. 

4 


26  ASTRONOMICAL  INSTRUMENTS. 

tions  upon  the  sun.  An  Apparent  Solar  Day  is  the  interval  be- 
tween two  successive  meridian  passages  of  the  sun's  centre  ;  com- 
mencing when  the  sun  is  on  the  superior  meridian.  It  appears 
from  observation  that  it  is  a  little  longer  than  a  sidereal  day,  and 
that  its  length  is  variable  during  the  year.  It  is  divided  into  24 
Apparent  Solar  Ifours. 

45.  Mean  Solar  Time  is  measured  by  the  diurnal  motion  of  an 
imaginary  sun,  called  the  Mean  Sun,  conceived  to  move  uniformly 
from  west  to  east  in  the  equator,  with  the  real  sun's  mean  motion 
in  the  ecliptic,  and  to  have  at  all  times  a  right  ascension  equal  to 
the  sun's  mean  longitude.    A  Mean  Solar  Day  commences  when 
the  mean  sun  is  on  the  superior  meridian,  and  is  divided  into  24 
Mean  Solar  Hours. 

Since  the  mean  sun  moves  uniformly  and  directly  towards  the 
east,  the  length  of  the  mean  solar  day  must  be  invariable. 

46.  The  Astronomical  Day  commences  at  noon,  and  is  divided 
into  24  hours  ;  but  the  Calendar  Day  commences  at  midnight,  and 
is  divided  into  two  portions  of  12  hours  each. 

47.  Astronomical  observations  are,  for  the  most  part,  made  in 
the  plane  of  the  meridian.     But  some  of  minor  importance  are 
made  out  of  this  plane.     The  chief  instruments  employed  for  me- 
ridian observations,  are  the  Astronomical  Circle,  and  the  Transit 
Instrument,   used  in  connection  with  the   Astronomical   Clock. 
These  are  the  capital  instruments  of  an  observatory,  inasmuch  as 
they  serve  (as  will  soon  be  explained)  for  the  determination  of  the 
places  of  the  heavenly  bodies,  which  are  the  fundamental  data  of 
astronomical  science.    The  principal  instruments  used  for  making 
observations  out  of  the  meridian  plane,  are  the  Altitude  and  Azi- 
muth Instrument,  the  Equatorial,  and  the  Sextant. 

TRANSIT  INSTRUMENT. 

48.  The  Transit  Instrument  is  a  meridional  instrument,  employ- 
ed in  conjunction  with  a  clock  or  chronometer  for  observing  the 
passage  of  celestial  objects  across  the  meridian,  either  for  the  pur- 
pose of  determining  their  difference  of  right  ascension,  or  obtaining 
the  correct  time.     It  is  constructed  of  various  dimensions,  from  a 
focal  length  of  20  inches  to  one  of  10  feet.     The  larger  and  more 
perfect  instruments  are  permanently  fixed  in  the  meridian  plane  ; 
the  smaller  ones  are  mounted  upon  portable  stands.     Fig.  15  rep- 
resents a  fixed  transit  instrument.     AD  is  a  telescope,  fixed,  as  it 
is  represented  in  the  figure,  to  a  horizontal  axis  formed   of  two 
cones.     The  two  small  ends  of  these  cones  are  ground  into  two 
perfectly  equal  cylinders  ;  which  cylindrical  ends  are  called  Pivots. 
These  pivots  rest  on  two  angular  bearings,  in  form  like  the  upper 
part  of  a  Y,  and  denominated  Y's.     The  Y's  are  placed  in  two 
dove-tailed  brass  grooves  fastened  in  two  stone  pillars  E  and  W,  so 
erected  as  to  be  perfectly  steady.    One  of  the  grooves  is  horizontal, 
the  other  vertical,  so  that,  by  means  of  screws,  one  end  of  the  axis 


TRANSIT    INSTRUMENT.  27 

may  be  pushed  a  little  forward  or  backward,  and  the  other  end 
may  be  either  slightly  depressed  or  elevated :  which  two  small 
movements  are  necessary,  as  it  will  be  soon  explained,  for  two  ad 
justments  of  the  telescope. 

Fig.  15. 


Let  E  be  called  the  eastern  pillar,  W  the  western.  On  the 
eastern  end  of  the  axis  is  fixed  (so  that  it  revolves  with  the  axis) 
an  index  n,  the  upper  part  of  which,  when  the  telescope  revolves, 
nearly  slides  along  the  graduated  face  of  a  circle,  attached,  as  it  is 
shown  in  the  figure,  to  the  eastern  pillar.  The  use  of  this  part  of 
the  apparatus  is  to  adjust  the  telescope  to  the  altitude  or  zenith 
distance  of  a  star  the  transit  of  which  is  to  be  observed.  Thus, 
suppose  the  index  n  to  be  at  o,  in  the  upper  part  of  the  circle, 
when  the  telescope  is  horizontal :  then,  by  elevating  the  telescope, 
the  index  is  moved  downward.  Suppose  the  position  to  be  that 
represented  in  the  figure,  then  the  number  of  degrees  between  o 
and  the  index  is  the  altitude. 

The  wire  plate  placed  in  the  focus  of  the  transit  telescope,  has 
attached  to  it  five  vertical  wires  together  with  one  horizontal  wire. 
In  order  to  be  seen  at  night,  these  wires,  or  rather  the  field  of  view, 
requires  to  be  illuminated  by  artificial  light.  The  illumination  of 
the  field  is  effected  by  making  one  of  the  cones  hollow,  and  ad- 
mitting the  light  of  a  lamp  placed  in  the  pillar  opposite  the  orifice ; 
which  light  is  directed  to  the  wires  by  a  reflector  placed  diagonally 
in  the  telescope.  The  reflector,  having  a  large  hole  in  its  centre, 


28  ASTRONOMICAL  INSTRUMENTS. 

does  not  interfere  with  the  rays  passing  down  the  telescope  from 
the  object* 

The  wires  are  seen  as  dark  lines  upon  a  bright  ground.  In 
some  of  the  best  instruments  recently  constructed  there  is  a  neat 
contrivance  for  illuminating  the  wires  directly,  so  as  to  make  them 
appear  bright  upon  a  dark  ground,  which  is  intended  to  be  used  in 
making  observations  upon  faint  stars. 

Sometimes  the  transit  instrument  is  furnished  with  a  meridian  graduated  circle 
of  large  size,  designed  to  be  used  for  the  measurement  of  meridian  altitudes  or 
zenith  distances.  It  then  takes  the  name  of  Meridian  Circle  or  Transit  Circle ; 
and  serves  for  the  determination  of  both  the  right  ascension  and  declination  of  a 
heavenly  body.  The  meridian  circle  of  the  observatory  recently  established  at 
Pulkova,  near  St.  Petersburg,  has  two  meridian  limbs,  provided  each  with  four 
reading  microscopes. 

49.  We  will  now  explain  the  principal  adjustments  of  the  tran- 
sit. Upon  setting  the  instrument  up  it  should  be  so  placed  that 
the  telescope,  when  turned  down  to  the  horizon,  should  point  north 
and  south,  as  near  as  can  possibly  be  ascertained.  This  being 
done,  then — 

(1.)  To  adjust  the  line  of  collimation. 

This  adjustment  consists  in  bringing  the  central  vertical  wire, 
within  the  telescope,  to  intersect  the  optical  axis,  which  is  sup- 
posed to  be  fixed  by  the  maker  of  the  instrument  perpendicularly 
to  the  axis  of  rotation.  There  is  no  occasion  with  this  instrument 
to  have  the  horizontal  wire  intersect  the  .  optical  axis  with  exact- 
ness. Direct  the  telescope  to  some  small,  distant,  well-defined 
object,  (the  more  distant  the  better,)  and  bisect  it  with  the  middle 
of  the  central  vertical  wire ;  then  lift  the  telescope  out  of  its 
angular  bearings,  or  Y's,  and  replace  it  with  the  axis  reversed. 
Point  the  telescope  again  to  the  same  object,  and  if  it  be  still  bi- 
sected, the  collimation  adjustment  is  correct;  if  not,  move  the 
wires  one  half  the  angle  of  deviation,  by  turning  the  small  screws 
that  hold  the  wire  plate,  near  the  eye-end  of  the  telescope,  and  the 
adjustment  will  be  accomplished  :  but,  as  half  the  deviation  may 
not  be  correctly  estimated  in  moving  the  wires,  it  becomes  neces- 
sary to  verify  the  adjustment  by  moving  the  telescope  the  other 
half,  which  is  done  by  turning  the  screw  that  gives  the  small  azi- 
muth motion  to  the  Y  before  spoken  of,  and  consequently  to  the 
pivot  of  the  axis  which  it  carries.  Having  thus  again  bisected  the 
object,  reverse  the  axis  as  before,  and  if  half  the  error  was  cor- 
rectly estimated,  the  object  will  be  bisected  upon  the  telescope 
being  directed  to  it.  If  it  should  not  be  bisected,  the  operation  of ' 
reversing  and  correcting  half  the  error  must  be  gone  through  again, 
and  until  after  successive  approximations  the  object  is  found  to  be 
bisected  in  both  positions  of  the  axis  ;  the  adjustment  will  then  be 
perfect.* 

*  Woodhouse's  Astronomy,  vol.  i.  pp.  70-72  ;  also  Simm's  Treatise  on  Mathe* 
matical  Instruments,  p.  53. 


TRANSIT    INSTRUMENT.  29 

It  is  desirable  that  the  central  wire  should  be  truly  vertical,  as 
we  should  then  have  the  power  of  observing  the  transit  of  a  star 
on  any  part  of  it,  as  well  as  the  centre.  It  may  be  ascertained 
whether  it  is  so,  by  elevating  and  depressing  the  telescope,  when 
directed  to  a  distant  object :  if  the  object  is  bisected  by  every  part 
of  the  wire,  the  wire  is  vertical,  (or  rather  it  is  perpendicular  to 
the  axis  of  rotation  of  the  telescope,  and  becomes  vertical  so  soon 
as  the  axis  of  rotation  is  made  horizontal.)  If  it  is  not  bisected, 
the  wire  should  be  adjusted,  by  turning  the  inner  tube  carrying 
the  wire  plate  until  the  above  test  of  its  vertically  be  obtained. 

50.  (2.)  To  set  the  axis  of  rotation  of  the  telescope  horizon- 
tal.  This  adjustment  is  effected  by  means  of  a  spirit-level ;  either 
attached  to  two  upright  arms  bent  at  their  upper  extremities,  by 
which  it  is  hung  on  the  two  pivots  of  the  axis,  or  else  having  two 
legs  and  standing  upon  the  axis.     In  the  first  position  it  is  called 
a  hanging  level,  and  in  the  second  a  riding  level.     At  one  end  of 
the  level  is  a  vertical  adjusting  screw,  by  which  that  end  may  be 
elevated  or  depressed.     Put  the  level  in  its  place,  and  observe  to 
which  end  of  the  level  the  bubble  runs,  which  will  always  be  the 
more  elevated  end ;  bring  it  back  to  the  middle  by  the  Y  screw  for 
vertical  motion,  and  take  off  the  level  and  hang  it  on  again  with 
the  ends  reversed.     Then,  if  the  bubble  is  again  found  in  the  mid- 
dle, the  level  is  already  parallel  to  the  axis,  and  the  axis  horizon- 
tal ;  but  if  not,  adjust  one  half  the  error  by  the  adjusting  screw  of 
the  level,  and  the  other  half  by  the  Y  screw ;  and  let  the  operation 
of  reversing,  and  adjusting  by  halves,  be  repeated  until  the  bubble 
will  remain  stationary  in  either  position  of  the  level,  in  which  case 
both  the  level  and  axis  will  be  horizontal. 

51 .  (3.)  To  adjust  the  line  of  collimation  to  the  plane  of  the  me- 
ridian.    We   have  said,  that  upon  setting  the  instrument  up,  the 
telescope  is  to  be  brought  into  the  meridian  plane,  as  near  as  can 
be  ascertained.     One  mode  of  establishing  it,  is  to  direct  the  tel- 
escope to  the  pole  star,  and  by  repeated  observations  find  the 
position  corresponding  to  its  greatest  or  least  altitude.    At  the 
present  time,  we  may  instead  compute  by  means  of  existing  tables 
founded  on  observation,  the  time  of  the  meridian  transit  of  the 
pole  star,  and  at  that  computed  time  bisect  the  star  by  the  middle 
vertical  wire.     Afterwards  the  line  of  collimation  may  be  placed 
still  more  exactly  in  the  plane  of  the  meridian  in  the  following 
manner :  Note  the  times  of  two  successive  superior  transits  of  the 
pole  star  across  the  central  vertical  wire,  and  the  time  of  the  inter- 
vening inferior  transit.     If  the  line  of  collimation  were  exactly  in 
the  plane  of  the  meridian,  as  the  diurnal  circles  are  bisected  by 
this  plane,  the  interval  between  the  superior  and  next  inferior  tran- 
sit would  be  precisely  equal  to  the  interval  between  the  inferior 
and  next  superior  transit.     Accordingly,  if  these  intervals  are  not 
in  fact  equal,  find  by  repeated  trials  the  position  of  the  telescope 


30  ASTRONOMICAL  INSTRUMENTS. 

and  vertical  wire  for  which  they  are  equal,  and  the  line  of  collima 
tion  will  then  be  in  the  plane  of  the  meridian. 

Instead  of  establishing  this  equality  by  a  system  of  trial  and 
error,  we  may,  by  means  of  a  formula  which  has  been  investiga- 
ted for  the  purpose,  compute  from  an  observed  inequality  the 
amount  of  the  movement  in  azimuth  necessary  to  correct  the  error 
of  position  of  the  instrument. 

Another,  and  generally  a  more  convenient  method,  is  to  observe  one  of  the  tran- 
sits of  the  pole  star,  and  also  the  transit  of  some  star  that  crosses  the  meridian  near 
the  zenith,  and  which  follows  or  precedes  the  pole  star  by  a  known  interval,  (differ- 
e.nce  of  right  ascensions  of  the  two  stars,)  and  compare  the  observed  interval  with 
the  calculated  interval.  The  difference  of  the  two  may  be  made  to  disappear  by 
repeated  trials :  or  a  formula  may  easily  be  investigated,  which  shall  make  known 
the  angular  movement  of  the  instrument  necessary  to  make  the  observed  and  cal- 
culated intervals  precisely  equal. 

The  method  of  regulating  the  clock  required  in  making  this  ad- 
justment, will  be  explained  when  we  come  to  treat  of  the  astro- 
nomical clock. 

52.  When  the  transit  telescope  has  once  been  placed  accurately 
in  the  meridian  plane,  in  order  to  avoid  the  repetition  of  trouble- 
some verifications  of  its  position,  a  meridian  mark  should  be  set 
up,  and  permanently  established,  at  a  distance  from  the  instru- 
ment ;  its  place  being  determined  by  means  of  the  middle  or  me- 
ridional wire.     At  Greenwich  two  such  marks,  one  to  the  north 
and  another  to  the  south,  are  used  ;  they  are  vertical  stripes  of 
white  paint  upon  a  black  ground,  on  buildings  about  two  miles  dis- 
tant from  the  observatory.  The  position  of  the  telescope  is  verified 
by  sighting  at  the  meridian  mark,  when  it  is  once  established. 

53.  The  times  of  the  transits  of  the  heavenly  bodies  are  ascer- 
tained as  follows :  in  the  case  of  a  star,  the  moments  of  its  cross- 
ing each  of  the  five  vertical  wires  are  noted ;  as  the  wires  are 
equally  distant  from  each  other,  the  mean  of  these  times  (or  their 
sum  divided  by  5)  will  be  the  time  of  the  star's  crossing  the  mid- 
dle wire,  or  of  its  meridian  transit.     The  utility  of  having  five 
wires,  instead  of  the  central  one  only,  will  be  readily  understood, 
from  the  consideration  that  a  mean  result  of  several  observations 
is  deserving  of  more  confidence  than  a  single  one ;   since  the 
chances  are  that  an  error  which  may  have  been  made  at  one  wire 
will  be  compensated  by  an  opposite  error  at  another.*     If  the  body 
observed  has  a  disc  of  perceptible  magnitude,  as  in  the  cases  of  the 
sun,  moon,  and  planets,  the  times  of  the  passage  of  both  the  west- 
ern and  eastern  limb  across  each  of  the  five  wires  are  noted,  and 
the  mean  of  the  whole  taken,  which  will  be  the  instant  of  the  me- 
ridian transit  of  the  centre  of  the  body. 

The  time  of  the  meridian  transit  of  a  body  may,  in  this  manner, 
be  ascertained  within  a  few  tenths  of  a  second. 

54.  When  a  star  is  on  the  meridian,  its  declination  circle  (Def 

*  Simm's  Mathematical  Instruments,  p.  59. 

'*«  * 


ASTRONOMICAL    CLOCK.  31 

16,  p.  15)  coincides  with  the  meridian;  moreover,  the  arc  of  the 
equator  which  lies  between  the  declination  circles  of  two  stars, 
measures  their  difference  of  right  ascension,  (see  def.  26,  p.  17.) 
It  follows,  therefore,  that  in  the  interval  between  the  transits  of  any 
two  stars,  the  arc  of  the  equator  which  expresses  their  difference 
of  right  ascension  will  pass  across  the  meridian,  the  rate  of  the 
motion  being  that  of  1 5°  to  a  sidereal  hour :  hence  the  difference 
of  the  times  of  transit  of  two  stars,  as  observed  with  a  sidereal 
clock,  when  converted  into  degrees  by  allowing  15a  to  the  hour, 
will  be  the  difference  between  the  right  ascensions  of  the  two  stars. 
We  may,  then,  in  this  manner,  by  means  of  a  transit  instrument 
and  sidereal  clock,  find  the  differences  between  the  right  ascension 
of  any  one  star  and  the  right  ascensions  of  all  the  others.  This 
being  done,  as  soon  as  the  position  of  the  vernal  equinox  with  re- 
spect to  the  same  star  becomes  known,  (and  we  shall  show  how 
to  find  it,)  the  absolute  right  ascensions  of  all  the  stars  will 
also  become  known.  Thus  RR',  (Fig.  8,)  is  the  difference  of 
right  ascension  of  the  stars  S  and  S',  their  absolute  right  as- 
censions being  VR  and  VR',  and  VR  is  the  distance  of  the  vernal 
equinox  V  from  the  declination  circle  of  the  star  S  ;  and  it  will  at 
once  be  seen  that  if  RR'  be  found,  in  the  manner  just  explained 
so  soon  as  VR  becomes  known,  by  adding  it  to  RR'  we  shall  have 
VR'  the  right  ascension  of  the  star  S'.  In  the  actually  existing 
state  of  astronomical  science,  the  right  ascensions  of  all  the  stars 
are  more  or  less  accurately  known,  and  a  right  ascension  sought 
is  now  obtained  directly,  by  noting  the  time  of  the  transit  of  the 
body  with  a  sidereal  clock  regulated  so  as  to  indicate  Oh.  Om.  Os. 
when  the  vernal  equinox  is  on  the  meridian,  and  converting  it  into 
degrees. 

ASTRONOMICAL  CLOCK. 

55.  The  astronomical  clock  is  very  similar  to  the  common  clock. 
It  has  a  compensation  pendulum  ;  that  is,  a  pendulum  so  construct- 
ed that  its  length  is  unaffected  by  changes  of  temperature.     The 
hours  on  the  face  are  marked  from  1  to  24. 

56.  Astronomers  make  use  of  sidereal  time  (as  already  stated) 
in  determining  the  right  ascensions  of  the  heavenly  bodies,  but  for 
all  other  purposes  they  generally  use  mean  solar  time. 

57.  To  regulate  a  sidereal  clock. — When  a  clock  is  used  for  de 
termining  differences  of  right  ascension,  (54,)  it  is  adjusted  to  side- 
real time  if  it  goes  equally  and  marks  out  24  hours  in  a  sidereal 
day ;  it  being  altogether  immaterial  at  what  time  it  indicates  Oh. 
Om.  Os.     To  ascertain  the  daily  rate  of  going  of  a  clock  which  is 
to  be  adjusted  to  sidereal  time  for  the  purpose  just  mentioned,  note 
by  the  clock  the  times  of  two  successive  meridian  transits  of  the 
same  star :  the  difference  between  the  interval  of  the  transits  and 
24  hours  will  be  the  daily  gain  or  loss  (as  the  case  may  be)  of  the 


32  ASTRONOMICAL  INSTRUMENTS. 

clock  with  respect  to  a  perfectly  accurate  sidereal  clock.*  If  the 
gain  or  loss,  when  found  after  this  manner,  proves  to  be  the  same 
each  day,  then  the  mean  rate  of  going  is  the  same  each  day. 

Next,  to  be  able  to  discover  the  rate  from  hour  to  hour  during  the  day,  it  is  ne. 
cessary  to  have  obtained  beforehand,  at  various  times,  and  under  various  states  of 
the  circumstances  likely  to  influence  the  rate  of  going  of  the  clock,  the  differences 
between  the  times  of  the  transits  of  a  number  of  different  stars,  (correcting  propor- 
tionally for  the  daily  rate,)  and  to  take  the  mean  of  the  several  differences  found  for 
each  pair  of  stars  for  the  exact  difference  of  their  transits.  When  this  lias  been 
done,  the  rate  of  the  clock  may  be  found  at  all  hours  during  the  day  by  noting  by 
the  clock  the  differences  between  the  times  of  the  transits  of  these  stars,  and  com- 
paring these  with  the  exact  differences  already  found.  At  the  present  time,  the 
right  ascensions  of  the  stars  being  known,  to  ascertain  the  rate  from  hour  to  hour, 
we  have  only  to  compare  the  intervals  of  time  given  by  the  clock  between  the 
transits  of  different  stars  taken  in  the  order  of  their  right  ascension  with  their 
differences  of  right  ascension. 

58.  The  sidereal  clocks  now  in  use  are  made  to  indicate  Oh, 
Om.  Os.  when  the  vernal  equinox  is  on  the  superior  meridian.    For 
the  regulation  of  such  clocks,  it  is  necessary  to  know  not  only  their 
rate  but  also  their  error.     This  is  found  by  noting  the  time  of  the 
transit  of  a  star,  and  comparing  this  with  its  right  ascension  ex- 
pressed in  time.    If  the  two  are  equal  the  clock  is  right,  otherwise 
their  difference  will  be  its  error. 

If  the  error  of  the  rate  of  a  clock  be  considerable,  it  should  be 
diminished  by  altering  the  length  of  the  pendulum  ;  otherwise,  it 
may  be  allowed  for.  The  stars  best  adapted  to  the  regulation  of 
clocks  are  those  in  the  vicinity  of  the  equator ;  for,  as  their  motion 
is  more  rapid  than  that  of  the  stars  more  distant  from  the  equator, 
there  is  less  liability  to  error  in  noting  their  transits. 

59.  A  mean  solar  clock  is  usually  regulated  by  observations  up- 
on the  sun.     The  method  of  regulating  it  cannot  be  adequately 
explained  until  we  have  treated  of  the  apparent  motion  of  the  sun. 
It  will  here  suffice  to  state,  that  with  the  instruments  we  have  now 
described,  the  sun's  motion  can  be  ascertained ;  and  therefore,  as 
a  knowledge  of  this  is  all  that  is  necessary  in  order  that  we  may 
be  able  to  obtain  the  mean  solar  time  at  any  instant,  that  it  is  pos- 
sible to  express  all  intervals  of  time  in  mean  solar  time. 

ASTRONOMICAL  CIRCLE. 

60.  An  Astronomical  Circle  is  an  instrument  designed  for  the 
measurement  of  the  zenith  distances  or  altitudes  of  the  heavenly 
bodies  at  the  instants  of  their  arrival  on  the  meridian.     Its  essen- 
tial parts  are  a  graduated  circular  limb,  a  telescope  turning  upon  a 
horizontal  axis  which  passes  through  the  centre  of  the  limb,  and  a 
micrometer  microscope,  (40,)  or  other  piece  of  apparatus,  for  read- 
ing off   the  angles  upon  the  limb.     It  is   sometimes   mounted 
upon  an  upright  stem,  which  either  turns  upon  fixed  supports  or 

*  It  is  not  necessary,  in  order  to  obtain  the  daily  rate  of  a  sidereal  clock,  that 
the  transit  instrument  should  be  adjusted  to  the  plane  of  the  meridian.  It  is  only 
requisite  that  it  should  be  kept  fixed  in  some  one  vertical  plane. 


MURAL  CIRCLE. 


33 


rests  upon  a  tripod,  and  can  be  turned  around  in  azimuth ;  but,  in 
general,  the  larger  circles  in  the  best  furnished  observatories  have 
their  axis  let  into  a  massive  pier,  or  wall,  of  stone,  and  capable 
of  only  such  small  motions  in  the  horizontal  and  vertical  directions, 
under  the  action  of  screws,  as  may  be  necessary  for  its  adjustment 
to  the  horizontal  position  and  perpendicular  to  the  meridian  plane. 
These  are  called  Mural  Circles.  For  greater  accuracy  the  angle 
is  read  off  at  six  different  points  of  the  limb  by  means  of  six  sta- 
tionary micrometer  microscopes,  and  the  mean  of  the  different 
readings  taken  for  the  angle  required.  Fig.  16  is  a  side  view  of  a 
mural  circle.  The  graduation  is  on  the  outer  rim,  which  is  per- 
pendicular to  the  plane  of  the  wall  CDFE.  One  of  the  reading 

Fig.  16. 


microscopes  is  represented  at  A.  The  others,  which  are  omitted 
in  the  figure,  are  disposed  at  equal  distances  around  the  rim.  The 
position  of  the  telescope  may  be  changed  by  unclamping  it  and 
clamping  it  to  a  different  part  of  the  limb.  In  taking  an  angle,  the 
telescope  is  made  fast  to  the  limb,  and  the  limb  glides  past  the  sta- 
tionary microscopes. 

The  six  reading  microscopes,  together  with  the  power  of  chang- 
ing the  position  of  the  telescope  on  the  limb,  so  as  to  read  off  the 
angle  from  all  parts  of  the  limb,  when  the  mean  results  of  a  great 
number  of  observations  are  taken,  do  away  with,  or  at  least  very 
considerably  lessen  the  errors  of  graduation,  centring,  and  une- 
qual expansion. 

o 


34  ASTRONOMICAL  INSTRUMENTS. 

61.  In  place  of  mural  circles,  Mural  Quadrants  have  been  much  used.     Since 
the  mural  quadrant  has  its  graduated  limb  only  one  fourth  the  size  of  the  limb  of 
the  mural  circle,  it  can  be  made  larger  than  the  circle.     But  the  circle  is  better 
balanced  than  the  quadrant,  and  the  quadrant  does  not  possess  the  advantages 
which  have  been  enumerated  as  resulting  in  the  case  of  the  circle  from  the  use  of 
a  number  of  reading  microscopes,  and  from  the  power  to  change  the  position  of 
the  telescope  on  the  limb.     Besides,  two  mural  quadrants,  one  to  observe  the  stars 
north  of  the  zenith,  and  another  to  observe  the  stars  south  of  the  zenith,  are  neces, 
sary  to  effect  the  general  object,  accomplished  by  one  mural  circle,  of  ascertaining 
the  zenith  distance  or  altitude  of  any  heavenly  body  at  the  time  of  its  arrival  on 
the  meridian. 

62.  The  largest  astronomical  circles  that  have  yet  been  con- 
structed, are  to  be  found,  it  is  said,  in  the  Dublin  and  Cambridge 
Observatories.    That  in  the  Dublin  Observatory  is  8  feet  in  diame- 
ter, and  has  an  azimuth  motion,  (that  is,  a  motion  about  a  vertical 
axis.)     The  other  is  a  mural  circle.     The  mural  circle  in  the  Na- 
tional Observatory  at  Washington  has  a  diameter  of  a  little  more 
than  5  feet. 

The  large  mural  quadrants  of  the  Greenwich  Observatory  are  of  8  feet  radius. 

63.  There  is  another  modification  of  the  astronomical  circle,  called  the  Zenith 
Sector,  which  is  used  to  measure  the  meridian  zenith  distances  of  stars  that  cross 
the  meridian  within  a  few  degrees  of  the  zenith.     The  limb  extends  only  about* 
10°  on  each  side  of  the  lowest  point.   It  can,  accordingly,  be  made  larger  than  the 
limb  of  the  circle  or  quadrant.    The  zenith  sector  in  the  observatory  at  Greenwich 
has  a  radius  of  12  feet. 

64.  The  mural  circle,  like  the  transit  instrument,  requires  three 
adjustments  :   1 .  Its  axis  must  be  made  horizontal ;  2.  Its  line  of 
collimation(38)  must  be  made  perpendicular  to  the  horizontal  axis; 
3.  The  line  of  collimation  must  be  made  to  move  in  the  plane  of 
the  meridian. 

A  simple  mechanical  contrivance  exists  for  carrying  the  first  of 
the  adjustments  into  complete  effect.  When  the  axis  is  made  ho- 
rizontal, the  line  of  collimation  describes  a  vertical  circle  ;  but  it 
may  describe  a  small  circle  of  the  celestial  sphere.  To  make  it  ne- 
cessarily describe  a  great  circle,  and  a  meridional  circle,  there  are 
no  mechanical  means.  Astronomical  ones  must  be  resorted  to ; 
and  ejen  with  those,  the  two  latter  adjustments  are  not  accom- 
plished without  great  difficulty,  We  may,  on  this  occasion,  use 
the  transit.  When  a  star  is  on  the  meridional  wire  of  the 
transit  instrument,  so  move  the  mural  circle  that  the  star  may 
be  on  its  middle  wire.  Next,  observe  by  the  transit  instrument 
when  a  star,  on,  or  very  near  to  the  zenith,  crosses  the  meridian : 
if,  at  that  time,  the  star  is  on  the  middle  vertical  wire  of  the  tele- 
scope of  the  mural  circle,  then  its  line  of  collimation  is  rightly 
adjusted.  If  the  star  is  on  the  middle  wires  of  the  two  telescopes 
at  different  times,  note  their  difference  and  adjust  accordingly.* 

65.  The  horizontal  point  of  the  limb,  technically  so  called,  is 
the  place  of  the  index  (or  centre  of  the  microscope)  answering  to 

*  This  adjustment  must  be  conducted  by  some  formula  which  expresses  the  re- 
lation between  the  difference  of  the  times,  and  the  inclination  of  the  line  of  collima- 
tion to  the  plane  of  the  meridian,  (Woodhouse's  Astronomy,  p.  117.) 


MURAL    CIRCLE.  3$ 

a  horizontal  position  of  the  line  of  collimation  of  the  telescope. 
Perhaps  the  simplest  method  of  obtaining  this  point  is  the  follow- 
ing :  Direct  the  telescope  upon  some  star  at  the  moment  of  its 
culmination,  and  read  off  the  angle  on  the  limb.  Procure  an  arti- 
ficial horizon,  (see  art.  79,)  and  on  the  following  night  direct  the 
telescope  upon  the  image  of  the  same  -star,  as  seen  in  the  artificial 
horizon.  By  the  laws  of  reflexion,  the  angle  of  depression  of  this 
image  will  be  equal  to  the  angle  of  elevation  of  the  star.  Accord- 
ingly the  arc  on  the  limb  which  passes  before  the  reading  micro- 
scope, in  moving  the  telescope  from  the  star  to  its  image,  will  be 
double  the  altitude  of  the  star,  and  its  point  of  bisection  the  hori- 
zontal point.*  This  point  may  also  be  found  by  directing  the 
telescope  upon  a  star  whose  altitude  is  known. 

66.  In  the  case  of  the  mural  quadrant,  if  there  is  no  altitude  that  can  be  relied 
on  as  having  been  obtained  with  all  attainable  accuracy,  it  is  necessary  to  have 
recourse  to  the  zenith  sector.  This  instrument  is  so  constructed  and  arranged, 
that  its  horizontal  axis  can  be  reversed  in  position.  By  taking  the  zenith  distance 
of  a  star  with  its  face  towards  the  east,  and  then  of  the  same  star  with  the  face  to- 
wards the  west,  the  half  sum  of  the  two  will  be  its  true  zenith  distance.  With  this 
we  may  readily  find  the  vertical  point,  and  thence  the  horizontal  point,  on  the  limb 
of  the  mural  quadrant,  by  directing  the  telescope  upon  the  star  observed  with  the 
sector,  when  it  is  on  the  meridian. 

67.  The  adjustments  of  the  mural  circle  having  all  been  effect- 
ed, and  the  horizontal  point  determined,  if  the  instrument  be  set  to 
this  point,  and  the  telescope  afterwards  directed  upon  any  star  in 
the  meridian,  the  arc  of  the  limb  that  passes  by  the  reading  mi- 
croscope will  be  the  altitude  of  the  star.     In  making  the  observa- 
tion the  telescope  must  be  brought  into  such  a  position  that  the 
star  will  be  bisected  by  the  horizontal  wire,  as  it  passes  through 
the  field  of  view.     The  altitude  of  the  sun,  moon,  or  any  planet, 
may  be  ascertained  by  measuring  the  altitudes  of  the  upper  and 
lower  limbs,  and  taking  their  half  sum  for  the  altitude  of  the  cen- 
tre :  or,  if  the  apparent  semidiameter  be  known,  by  adding  this  to 
the  altitude  of  the  lower  limb,  or  subtracting  it  from  the  altitude  of 
the  upper  limb. 

68.  The  meridian  altitude  or  zenith  distance  of  a  heavenly  body 
having  been  measured  with  an  astronomical  circle,  or  other  similar 
instrument,  at  a  place  the  latitude  of  which  is  known,  its  declina- 
tion may  easily  be  found.     For,  let  s,  (Fig.  10,)  represent  the  point 
of  meridian  passage  of  a  star,  or  other  heavenly  body,  which  crosses 
the  meridian  to  the  north  of  the  zenith  (Z.)  Es  will  be  its  decima- 
tion, (Def.  27,  p.  17,)  Zs  its  meridian  zenith  distance,  and  ZE  the 
latitude  of  the  place  of  observation  (O,)  (Def.  33,  p.  18  :)  and  we 
obviously  have 

Es  =  ZE  +  Zs  .  .  .  (a). 

*  The  method  of  using  the  level  for  the  determination  of  the  horizontal  point 
may  be  found  explained  in  Herschel's  Astronomy,  p.  93.  Another  piece  of  appa- 
ratus, used  for  the  same  purpose,  called  the  Floating  Collimator,  is  described  in  the 
same  work,  p.  95. 


36  ASTRONOMICAL  INSTRUMENTS 

If  the  star  cross  the  meridian  at  some  point  s'  between  the  ze- 
nith (Z)  and  the  equator  (E,)  we  shall  have  Es'  =  ZE — Zs',  (b) , 
and  if  its  point  cf  transit  be  some  point  s"  to  the  south  of  the  equa- 
tor (E,)  we  shall  have  Es"  =  Zs"  —  ZE,  and  —  Es"  =  ZE  — 
Zs",  (c).  The  three  formulae  (a),  (6),  and  (c),  may  all  be  compre- 
hended in  one,  viz  : 

Declination  =  latitude  +  meridian  zenith  distance  .  .  .  ( 1 ) 
if  we  adopt  the  following  conventional  rules  :  1.  North  latitude  is 
always  positive ;  2.  The  zenith  distance  is  positive  when  it  is 
North,  that  is,  when  the  star  is  north  of  the  zenith,  and  negative 
when  it  is  South ;  3.  The  declination  is  North  if  it  comes  out  posi- 
tive, and  South  if  it  comes  out  negative. 

If  the  latitude  is  South,  it  must  be  regarded  as  negative,  and  the  zenith  distance 
must  be  affected  with  the  minus  sign  when  it  is  South,  and  with  the  plus  sign  when 
it  is  North.  The  rule  for  the  declination  is  the  same.  In  general,  North  latitude  is 
-f-,  South  latitude  — .  The  zenith  distance  has  the  same  sign  as  the  latitude  when 
it  is  of  the  same  name,  the  contrary  sign  when  it  is  of  a  contrary  name.  North 
declination  is  -}-,  South  declination  — . 

The  latitude  which  is  here  supposed  to  be  known,  maybe  found 
by  measuring  (67)  the  meridian  altitudes  of  a  circumpolar  star  at 
its  inferior  and  superior  transits,  and  taking  their  half  sum.  For, 
as  the  pole  lies  midway  between  the  points  at  which  the  transits 
take  place,  its  altitude  will  be  the  arithmetical  mean,  or  the  half 
sum  of  the  altitudes  of  these  points,  and  the  altitude  of  the  pole  is 
equal  to  the  latitude  of  the  place,  (34.) 

69.  When  the  right  ascension  and  declination  of  a  heavenly  body 
have  been  obtained  from  observation,  with  a  transit  instrument  and 
circle,  (54,  68,)  its  longitude  and  latitude  may  be  computed.    For, 
let  S  (Fig.  8)  represent  the  place  of  the  body,  VRQE  the  equa- 
tor, VLTW  the  ecliptic,  and  P,  K,  the  north  poles  of  the  equator 
and  ecliptic.     In  the  spherical  triangle  PKS  we  shall  know  PS 
the  complement  of  SR  the  declination,  and  the  angle  KPS  = 
ER  =  E  V  +  VR  =  90°  +  right  ascension ;  and  if  we  suppose  the 
obliquity  of  the  ecliptic  to  be  known,  we  shall  know  PK.     We 
may  therefore  compute  KS,  and  the  angle  PKS.     But  KS  is  the 
complement  of  SL,  which  is  the  latitude  of  the  body  S  ;   and 
PKS  =  180°  —  EKS  =  180°  —  (WV  +  VL)  =  180°  —  (90°  + 
longitude)  =  90°  —  longitude. 

The  obliquity  of  the  ecliptic,  which  we  have  here  supposed  to 
be  known,  is,  in  practice,  easily  found ;  for  it  is  equal  to  TQ,  the 
sun's  greatest  declination. 

ALTITUDE  AND  AZIMUTH  INSTRUMENT. 

70.  The  Altitude  and  Azimuth  Instrument  consists,  essentially, 
of  a  telescope  with  two  graduated  limbs,  the  one  horizontal  and  the 
other  vertical.     The  telescope  turns  about  the  centre  of  the  verti- 
cal limb,  or  turns  with  the  limb  about  its  centre ;  and  the  ver- 
tical limb  turns,  with  the  telescope,  about  the  vertical  axis  of  the 
horizontal  limb. 


EQUATORIAL. 


37 


If  the  telescope  be  brought  into  the  meridian  plane,  and  after- 
wards  directed  upon  a  star  out  of  this  plane,  the  arc  of  the  hori- 
zontal limb  passed  over  by  the  index  will  be  the  azimuth  of  the 
star.  The  vertical  limb  will  serve  to  measure  its  altitude, 

71.  The  Meridian  Line  (Def.  8,  p  14)  at  a  place  may  easily  be 
determined  with  the  altitude  and  azimuth  instrument,  by  a  method 


Fig.  17. 


called  the  Method  of  Equal  Alti- 
tudes. Let  O  (Fig.  17)  represent 
the  place  of  observation,  NPZ  the 
meridian,  and  S,  S'  two  positions 
of  the  same  star,  at  which  the  alti- 
tude is  the  same.  Now,  the  spher- 
ical triangles  ZPS  and  ZPS'  have 
the  side  ZP  common,  ZS=ZS7, 
and  (allowing  the  stars  to  move  in 
circles)  PS=PS'.  Hence  they  are 
equal,  and  consequently  the  angle 
PZS=PZS' ;  that  is,  equal  altitudes  of  a  star  correspond  to  equal 
azimuths.  Therefore,  by  bisecting  the  arc  of  the  horizontal  limb, 
comprehended  between  two  positions  of  the  vertical  limb  for  which 
the  observed  altitude  of  a  star  is  the  same,  we  shall  obtain  the  me- 
ridian line. 

The  meridian  line  may  be  approximately  determined  by  this  method  with  the 
common  theodolite  ;  the  observations  being  made  upon  the  sun.  The  result  will 
be  more  accurate  if  they  be  made  towards  the  summer  or  winter  solstice,  when  the 
sun  will  have  but  a  slight  motion  towards  the  north  or  south  in  the  interval  of  the 
observations.  It  is,  however,  easy  to  determine  and  allow  for  the  effect  of  the  sun's 
change  of  place  in  the  heavens. 

72.  When  the  time  is  accurately  known,  the  north  and  south  line  maybe  found 
very  easily  by  directing  the  telescope  of  any  instrument  that  has  a  motion  in  azi- 
muth upon  a  star  in  the  vicinity  of  the  pole  and  at  a  distance  from  the  zenith,  at 
the  moment  of  its  arrival  on  the  meridian,  (which,  as  will  be  understood  in  the  se- 
quel, can  now  easily  be  determined  from  existing  data.) 

EQUATORIAL. 

73.  The  Equatorial  is  similar,  in  its  construction,  to  the  altitude 
and  azimuth  instrument.     It  is  so  called  from  the  circumstance  of 
one  of  the  limbs  being  placed  in  a  position  parallel  to  the  plane  of 
the  equator.    The  axis  of  this  limb  is  then  parallel  to  the  axis  of 
the  heavens  ;  and  the  other  limb,  to  the  centre  of  which  the  tele- 
scope is  attached,  is  parallel  in  every  one  of  its  positions  to  the 
plane  of  some  one  celestial  meridian.     The  limb  which  is  parallel 
to  the  equator  serves  for  the  measurement  of  differences  of  right 
ascension,  and  the  other  for  the  measurement  of  declinations.  The 
equatorial  is  regarded  as  one  of  the  most  indispensable  instruments 
of  an  astronomical  observatory.     It  is  particularly  useful  in  the 
measurement  of  apparent  diameters,  and  in  all  observations  that 
require  the  telescope  to  be  directed  upon  a  body  for  a  considerable 
period  of  time  ;  as,  by  giving  the  limb  to  which  the  telescope  is 
attached  a  slow  motion  from  east  to  west,  the  body  may  be  follow 


ASTRONOMICAL  INSTRUMENTS, 


ed  in  its  diurnal  motion,  and  kept  continually  within  the  field  ot 
view.  This  motion  is  generally  produced  by  clock-work,  without 
the  use  of  the  hand. 

It  is  also  frequently  used  for  determining  the  right  ascension  and  declination  of  a 
comet,  or  other  heavenly  body,  which  for  some  reason  cannot,  at  the  time,  be  ob- 
served in  the  meridian  ;  and  for  finding  and  obtaining  a  protracted  view,  or  fixing 
more  accurately  the  place  of  an  object  invisible  to  the  naked  eye,  whose  place  has 
been  approximately  calculated  from  the  results  of  previous  observations.  Another 
important  object  to  which  it  may  be  applied,  is  the  determination  of  small  differ- 
ences of  right  ascension  and  declination,  and  thus  of  the  relative  positions  of  con- 
tiguous objects.  Its  determinations  of  declinations  and  differences  of  right  ascen- 
sion, in  general,  are  to  be  deemed  less  accurate  than  those  effected  with  the  mural 
circle  and  transit  instrument ;  as,  from  its  more  complicated  structure,  and  peculiar 
position,  the  parts  have  less  stability  and  are  more  subject  to  unequal  strains, 
bendings,  and  expansions,  than  those  of  the  instruments  just  named. 

74.  The  adjustments  of  the  equatorial  are  somewhat  complica- 
ted and  difficult.  They  are  best  performed  by  following  the  pole- 
star  round  its  entire  diurnal  circle,  and  by  observing,  at  proper 
intervals,  other  considerable  stars  whose  places  are  well  ascertained. 
(Herschel.) 


Fig.  18. 


75.  In  addition  to  the  instruments  that  have  now  been  de- 
scribed, which  are  designed  and  used  for  the  measurement 
of  the  angular  distances  of  bodies  from  some  fixed  point  or 
circle  in  the  heavens,  astronomers  have  found  it  convenient 
and  important  to  have  another  instrument,  or  piece  of  appa- 
ratus, with  which  to  determine  directly  the  relative  situation 
of  two  stars  that  are  near  to  each  other ;  so  near  as  to  be 
seen,  at  the  same  time,  in  the  same  field  of  view.  The  ap- 
paratus used  for  this  purpose  is  attached  to  the  telescope  of 
the  equatorial,  or  other  instrument,  and  is  called  a  Micrometer. 
Another  important  use  to  which  it  is  put,  is  the  measurement 
of  the  apparent  diameters  of  the  heavenly  bodies.  It  has  a 
variety  of  forms.  The  simplest  is  known  by  the  name  of  the 
Wire-Micrometer.  It  is  placed  in  the  focus  of  the  telescope. 
It  consists  of  two  forks  of  brass,  bb'b,  cc'c,  (Fig.  18)  sliding 
one  within  the  other,  and  having  each  a  very  fine  wire,  or 
spider-line,  e,  and  d,  stretched  perpendicularly  across  from 
one  prong  to  the  other.  These  forks  are  placed  length- 
wise in  a  shallow  rectangular  box,  aa'aa',  about  2  inches 
wide  and  4  inches  long ;  and  have  each  fine-threaded  mi- 
crometer screws,  /,  /,  working  against  the  ends,  b',  and  c'. 
The  graduated  heads  of  these  screws  are  not  represented  in 
the  figure,  but  they  may  be  seen  in  Fig.  19.  They  pass 
p.  .Q  through  the  ends  a',  a',  of  the  box, 

and  have  their  graduated  heads  on 
the  outside  of  it.  Between  the  ends 
b',  c'  of  the  two  forks  and  the  con- 
tiguous ends  a',  a'  of  the  box  are  two 
spiral  springs,  h,  h,  which  keep  the 
ends  of  the  forks  firmly  pressed 
against  the  ends  of  the  screws,  and 
draw  the  forks  outward  and  the 
wires  further  apart  whenever  the 
screws  are  loosened.  By  turning 
the  screws  in  the  opposite  direction 
the  forks  are  pushed  forward,  and 
the  wires  brought  nearer  to  each 
other.  The  number  of  complete  turns  and  parts  of  a  turn  made  by  each  screw,  as 


a'nri 


SEXTANT.  39 

shown  by  its  graduated  head,  will  make  known  the  fraction  of  an  inch  through 
which  the  end  of  it  and  the  contiguous  wire  is  moved.  The  screws  can  be  so  del- 
icately  cut  that  they  will  measure  with  accuracy  the  ym^  of  an  inch.  The  linear 
space  thus  measured  in  the  focus  of  the  telescope  must  be  converted  into  the  equiv- 
alent angular  space  in  the  heavens.  This  is  effected  by  fixing  upon  two  contigu- 
ous stars,  whose  distance  is  accurately  known,  and  measuring  with  the  micrometer 
the  linear  distance  of  their  images  formed  in  the  focus.  In  this  way  will  be  found 
how  many  seconds  of  angular  space  correspond  to  a  given  movement  of  either  of 
the  wires,  as  measured  by  the  micrometer  scale.  The  micrometer  box  is  fastened 
perpendicularly  across  the  eye-end  of  the  tube  of  the  telescope.  The  eye-piece 
of  the  telescope  screws  into  the  outer  face  of  the  box,  (see  Fig.  19,)  and  on  looking 
into  it,  the  wires  d,  e  within  the  box  are  seen  in  its  focus ;  where  also  the  images 
of  the  stars,  formed  by  the  object-glass,  fall.  To  save  the  necessity  of  counting 
the  revolutions  of  the  micrometer  screws,  a  linear  scale  is  placed  within  the  box, 
and  at  one  side  of  it,  consisting  of  a  series  of  teeth,  with  intervening  notches.  This 
is  represented  in  the  diagram,  (Fig.  18.)  A  motion  of  the  wire  from  one  notch  to 
another  answers,  say,  to  one  turn  of  the  screw,  and  to  1'  in  space. 

To  measure  the  angular  distance  of  two  stars,  the  wires  are  both  brought  into 
coincidence  at  the  zero  of  this  scale,  when  we  will  suppose  that  they  fall  between 
the  stars.  By  turning  the  screws  they  are  moved  from  this  position,  and  the  mo- 
tion is  continued  until  the  one  star  is  accurately  bisected  by  one  wire,  and  the  other 
star  by  the  other  wire.  The  number  of  notches  which  the  wires  have  passed  will 
express  the  number  of  minutes  in  the  space  between  the  stars ;  to  these  are  to  be 
added  the  seconds  answering  to  the  fractional  parts  of  a  revolution,  as  shown  by 
the  divided  heads  of  the  screws.  It  will  be  seen,  that  in  order  to  obtain  the  real 
distance  between  the  two  stars,  the  two  wires  d  and  e  must  be  brought  into  such 
a  position  as  to  be  perpendicular  to  the  line  of  the  stars.  This  is  effected  by  giving 
to  the  whole  box  a  revolving  motion  about  the  optical  axis  of  the  telescope,  and 
bringing  the  wire  /,  which  is  perpendicular  to  d  and  e,  into  such  a  position  as  to 
bisect  both  the  stars.  The  diameter  of  a  heavenly  body  is  measured  in  a  similar 
manner  ;  the  wires  being  brought  into  contact  with  the  opposite  limbs. 

76.  To  measure  the  angle  made  by  the  line  of  direction  of  two  stars  with  a  fixed 
line  passing  through  one  of  them,  it  is  necessary  that  the  micrometer  box  should 
not  only  have  a  revolving  motion  around  the  axis  of  the  telescope,  but  also  a  grad- 
uated circle  to  measure  its  amount.  The  cross-wire  I  is  brought  by  this  motion 
into  coincidence — first  with  one  line  and  then  with  the  other,  and  then  the  an- 
gle read  off.  In  this  way  may  be  found  the  angle  made  by  the  line  of  direction  of 
two  contiguous  stars  with  the  meridian,  or  a  line  perpendicular  to  the  meridian, 
at  the  moment  one  of  them  is  crossing  this  circle.  This  angle  is  called  the  Angle 
of  Position  of  the  two  stars,  and  the  micrometer  that  serves  to  measure  it  is  called 
a  Position  Micrometer.  The  position  of  the  wire  I  when  perpendicular  to  the  meri- 
dian may  be  found  by  turning  it  until  one  of  the  stars  runs  along  the  wire,  while 
the  telescope  of  the  equatorial  is  stationary.  Fig.  19  represents  a  position  microm- 
eter. The  micrometer  box  b,  with  its  attached  eye-piece  e,  is  connected  with  the 
circle  a,  and  is  turned  around  with  it  by  the  small  milled-head  screw  s,  which 
works  on  an  interior  toothed  wheel,  and  the  angle  is  read  off  upon  the  stationary 
graduated  circle  above  a,  by  aid  of  the  vernier,  moveable  with  the  plate  a 

SEXTANT. 

77.  The  instruments  which  have  now  been  described  are  ob- 
servatory instruments,  the  chief  design  of  whose  construction  is  tc 
furnish  the  places  of  the  heavenly  bodies  with  all  attainable  exact- 
ness. That  of  which  we  are  now  to  treat  is  much  less  exact, 
though  still  of  great  utility  in  determining  the  essential  data  of 
some  of  the  practical  applications  of  astronomical  science ;  as 
finding  the  latitude  and  longitude  of  a  place,  and  the  time  of  day: 
and  is  used  chiefly  by  navigators,  and  astronomical  observers  on 
land,  who  are  precluded,  by  their  situation  or  other  circumstances, 


40 


ASTRONOMICAL  INSTRUMENTS. 


Fig.  20. 


from  using  the  more  accurate  instruments  of  an  observatory.  It  is 
much  more  conveniently  portable  than  any  of  these,  and  has  not  to 
be  set  up  and  adjusted  at  every  new  place  of  observation.  Besides, 
as  it  is  held  in  the  hand,  it  can  be  used  at  sea,  where,  by  reason 
of  the  agitations  of  the  vessel,  no  instrument  supported  in  the  or- 
dinary way  is  of  any  service. 

78.  The  sextant  may  be  defined,  in  general  terms,  to  be  an  in- 
strument which  serves  for  the  direct  admeasurement  of  the  angu- 
lar distance  between  any  two  visible  points.  The  particular  quan- 
tities that  may  be  measured  with  it,  are,  1st,  the  altitude  of  .1 
heavenly  body  ;  2d,  the  angular  distance  between  any  two  visible 
objects  in  the  heavens,  or  on  the  earth.  Its  essential  parts  are  a 
graduated  limb  BC,  (Fig.  20,)  comprising  about  60  degrees  of  the 
entire  circle,  which  is  attached  to  a  triangular  frame  BAG  ;  two 
mirrors,  of  which  one  (A)  called  the  Index  Glass,  is  moveable  in 
c(  .mection  with  an  index  G  about  A  the  centre  of  the  limb,  and 

the  other  (D)  called  the  Ho- 
rizon Glass,  is  permanently 
fixed  parallel  to  the  radius 
AC  drawn  to  the  zero  point 
of  the  limb,  and  is  only  half- 
silvered,  (the  upper  half  be- 
ing transparent ;)  and  an  im- 
moveable  telescope  at  E, 
directed  towards  the  horizon- 
glass.  The  principle  of  the 
construction  and  use  of  the 
sextant  may  be  understood 
from  what  follows  :  A  ray  of 
light  SA  from  a  celestial  ob- 
ject S,  which  impinges  against 
the  index-glass,  is  reflected 
off  at  an  equal  angle,  and 
striking  the  horizon-glass  (D) 
is  again  reflected  to  E,  where  the  eye  likewise  receives  through 
the  transparent  part  of  that  glass  a  direct  ray  from  another  point 
or  object  S'.  Now,  if  AS'  be  drawn,  directed  to  the  object  S', 
SAS',  the  angular  distance  between  the  two  objects  S  and  S',  is 
equal  to  double  the  angle  CAG  measured  upon  the  limb  of  the 
instrument,  (AC  being  parallel  to  the  horizon-glass.)  For,  when 
the  index-glass  is  parallel  to  the  horizon-glass,  and  the  angle  on  the 
limb  is  zero,  AD,  the  course  of  the  first  reflected  ray,  will  make 
equal  angles  with  the  two  glasses,  and  therefore  the  angle  SAD 
will  become  the  angle  S'AD,  (=  ADE  ;)  and  the  observer,  look 
ing  through  the  telescope,  will  see  the  same  object  S'  both  b} 
direct  and  reflected  light.  Now,  if  the  index-glass  be  moved  from 
this  position  through  any  angle  CAG,  the  angle  made  by  the  re- 
flected ray  which  follows  the  direction  AD  with  this  glass,  will  be 


SEXTANT.  41 

diminished  by  an  amount  equal  to  this  angle  ;  for,  we  have  DAG  = 
DAC  —  CAG.  Therefore  the  angle  made  with  the  index-glass 
by  the  new  incident  ray  SA,  which  after  reflexion  now  pursues  the 
same  course  ADE,  and  reaches  the  eye  at  E,  as  it  is  always  equal 
to  that  made  by  the  reflected  ray,  will  be  diminished  by  this  amount. 
Consequently,  the  incident  ray  in  question  will,  on  the  whole,  that 
is,  by  the  diminution  of  its  inclination  to  the  mirror  by  the  angle 
CAG  and  by  the  motion  of  the  mirror  through  the  same  angle,  be 
displaced  towards  the  right,  or  upward,  an  angle  S'AS  equal  to 
2GAC.  Thus,  the  angular  distance  SAS'  of  two  objects  S,  S', 
seen  in  contact,  the  one  (S')  directly,  and  the  other  (S)  by  reflex- 
ion from  the  two  mirrors,  is  equal  to  twice  the  angle  CAG  that 
the  index-glass  is  moved  from  the  position  (AC)  of  parallelism  to 
the  horizon-glass. 

Hence  the  limb  is  divided  into  1 20  equal  parts,  which  are  called 
degrees  ;  and  to  obtain  the  angular  distance  between  two  points, 
it  is  only  necessary  to  sight  directly  at  one  of  them,  and  then 
move  the  index  until  the  reflected  image  of  the  other  is  brought 
into  contact  with  it ;  the  angle  read  off  on  the  limb  will  be  the  an- 
gle sought. 

To  obtain  the  angular  distance  between  two  bodies  which 
have  a  sensible  diameter,  bring  the  nearest  limbs  into  contact,  and 
to  the  angle  read  off  on  the  limb  add  the  sum  of  the  apparent  semi- 
diameters  of  the  two  bodies,  or  bring  the  farthest  limbs  into  con- 
tact, and  subtract  this  sum. 

79.  The  sextant  is  also  employed  to  take  the  altitude  of  a  heav- 
enly body.     A  horizontal  reflector,  called  an  Artificial  Horizon, 
is  placed  in  front  of  the  observer :  the  angle  between  the  body  and 
its  reflected  image  is  then  measured,  as  if  this  image  were  a  real 
object ;  the  half  of  which  will  be  the  altitude  of  the  body. 

A  shallow  vessel  of  mercury  forms  a  very  good  artificial  horizon. 

In  obtaining  tne  altitude  of  a  body,  at  sea,  its  altitude  above 
the  visible  horizon  is  measured,  by  bringing  the  lower  limb  into 
contact  with  the  horizon.  To  this  angle  is  added  the  apparent 
semi-diameter  of  the  body,  and  from  the  result  is  subtracted  the 
depression  of  the  visible  horizon  below  the  horizontal  line,  called 
the  Dip  of  the  Horizon. 

80.  Hadley's   Quadrant  differs  from  the  sextant  in  having  a 
graduated  limb  of  45°,  instead  of  60°,  in  real  extent,  and  a  sight 
vane  instead  of  a  small  telescope.     It  is  not  capable,  then,  of  meas- 
uring any  angle  greater  than  about  90°,  while  the  sextant  will 
measure  an  angle  as  great  as  120°,  or  even  140°,  (for  the  gradua- 
tion generally  extends  to  140°.)     The  quadrant  is  also  inferior  to 
ihe  sextant  in  respect  to  materials  and  workmanship,  and  its  meas- 
urements are  less  accurate. 

6 


42  ASTRONOMICAL  INSTRUMENTS. 


ERRORS  OF  INSTRUMENTAL  ADMEASUREMENT. 

81.  Whatever  precautions  may  be  taken,  the  results  of  instru- 
mental admeasurement  will  never  be  wholly  free  from  errors. 
Errors  that  arise  from  inaccuracy  in  the  workmanship  or  adjust- 
ment of  the  instrument  may  be  detected  and  allowed  for.  But 
errors  of  observation  are  obviously  undiscoverable.  Since,  how- 
ever, the  chances  are  that  an  error  committed  at  one  observation 
will  be  compensated  by  an  opposite  error  at  another,  it  is  to  be 
expected  that  a  more  accurate  result  will  be  obtained  if  a  great 
number  of  observations,  under  varied  circumstances,  be  made, 
instead  of  one,  and  the  mean  of  the  whole  taken  for  the  element 
sought.  And  accordingly,  it  is  the  uniform  practice  of  astronomi- 
cal observers  to  multiply  observations  as  much  as  is  practicable. 

TELESCOPE. 

82.  An  observatory  is  not  completely  furnished  unless  it  is  supplied  with  a  large 
telescope  for  examining  the  various  classes  of  objects  in  the  heavens,  and  one  or 
more  smaller  ones  for  exploring  the  heavens  and  searching  for  particular  objects 
invisible  to  the  naked  eye,  as  faint  comets,   and  making  observations  upon  occa- 
sional celestial  phenomena,  as  eclipses  of  the  sun  and  moon,  occultations  of  the 
stars,  &c.     Telescopes  are  divided  into  the  two  classes  of  Reflecting  and  Refract- 
ing Telescopes.     In  the  former  class  the  image  of  the  object  is  formed  by  a  con- 
cave speculum,  and  in  the  latter  by  a  converging  achromatic  lens.     This  image  is 
viewed  and  magnified  by  an  eye-glass,    or  rather  by  an  achromatic  eye-piece 
consisting  of  two  glasses.     In  the  simplest  form  of  the  reflecting  telescope,  the 
Herschelian,  the  image  formed  by  the  concave  speculum  is  thrown  a  little  to  one 
side,  and  near  the  open  mouth  of  the  tube,  where  the  observer  views  it  through  the 
eye-glass,  with  his  back  turned  towards  the  object. 

83.  The  magnifying  power  of  a  telescope  is  to  be  carefully  distinguished  from  its 
illuminating  and  space-penetrating  power.     A  telescope  magnifies  by  increasing 
the  angle  under  which  the  object  is  viewed :  it  increases  the  light  received  from 
objects,  and  reveals  to  the  sight  faint  stars,  nebulae,  &c.,  by  intercepting  and  con- 
verging to  a  point  a  much  larger  beam  of  rays.     The  magnifying  power  is  meas- 
ured by  the  ratio  of  the  focal  length  of  the  object-glass,  or  speculum,  to  that  of  the 
eye-piece.     The  illuminating  and  space-penetrating  power,  (for  faint  objects,)  if 
we  leave  out  of  view  the  amount  of  light  lost  by  reflexion  and  absorption,  is  meas- 
ured by  the  proportion  which  the  aperture  of  the  object-glass  or  speculum  bears  to 
the  pupil  of  the  eye.     Telescopes  are  provided  with  several  eye-glasses  of  various 
powers.     The  power  to  be  used  varies  with  the  object  to  be  viewed,  and  the  purity 
and  degree  of  tranquillity  of  the  atmosphere.     Of  two  telescopes  of  the  same  focal 
length,  that  which  has  the  largest  aperture  will  form  the  brightest  image  in  the 
focus,  and  therefore,  other  things  being  equal,  admit  of  the  use  of  the  most  power- 
ful eye-piece.     In  this  way  it  happens  that  the  available  magnifying  power  indi- 
rectly depends  materially  upon  the  size  of  the  aperture.     In  all  telescopes  there- is 
a  certain  fixed  ratio  between  the  aperture  and  focal  length,  or  at  least  limit  to  this 
ratio.    In  reflecting  telescopes  it  is  about  one  inch  of  aperture  for  every  foot  of  focal 
length,  and  in  refracting,  one  inch  of  aperture  for  from  one  to  two  feet  of  focal 
length.     Reflectors  and  refractors  of  the  same  focal  length  have  about  the  same 
actual  magnifying  and  illuminating  power.     The  highest  available  magnifying 
power  that  has  yet  been  obtained  is  about  6,000  ;  but  this  was  applicable  only  to 
the  faintest  stars  and  nebulous  spots.  With  the  best  telescopes  a  magnifying  power 
of  a  few  hundred  is  the  highest  that  can  be  applied  to  the  moon  and  planets.     The 
largest  reflecting  telescope  that  has  yet  been  constructed,  and  directed  to  the  heav- 
ens, is  the  celebrated  one  of  Sir  William  Herschel,  of  40  feet  focus,  and  4  feet 
aperture.     Its  illuminating  power  was  about  35,000,  which  makes  its  space-pene 


CORRECTIONS    OF    CO-ORDINATES.  43 

trating  power  nearly  190  times  the  distance  of  the  faintest  star  visible  to  the  naked 
eye ;  and  its  highest  magnifying  power  was  6,450.*  The  most  powerful  refractor 
yet  constructed  is  in  the  new  observatory  at  Pulkova,  near  St.  Petersburg.  It  has 
an  aperture  of  very  nearly  15  inches,  (14.93  inches,)  and  a  focal  length  of  22  feet. 
The  best  telescope  in  the  United  States  is  the  refractor  in  the  new  observatory  at 
Cincinnati.!  Its  aperture  is  12  inches,  and  focal  length  about  17  feet.  The 
field  of  view  of  telescopes  diminishes  in  proportion  as  the  magnifying  power  in- 
creases. It  's  stated  that  with  a  magnifying  power  of  between  100  and  200  it  is 
a  circle  not  as  large  as  the  full  moon ;  and  with  a  power  of  600  or  1000  is  nearly 
filled  by  one  of  the  planets,  while  a  star  will  pass  across  it  in  from  two  to  three 
seconds. 

84.  The  diminution  of  the  field  of  view,  and  the  trepidations  of  the  image  oc- 
casioned by  the  varying  density  of  the  atmosphere,  and  the  unavoidable  tremors  of 
the  instrument,  must  ever  affix  a  practical  limit  to  the  magnifying  power  of  tele- 
scopes.    This  limit,  it  is  probable,  is  already  nearly  attained,  for  the  highest  pow- 
ers of  the  best  telescopes  can  now  be  used  only  in  the  most  favorable  states  of  the 
weather4 

85.  The  largo  refracting  telescopes  are  equatorially  mounted,  that  they  may,  as 
readily  as  possible,  be  directed  and  retained  upon  an  object. 

86.  The  small  telescope,  called  a  comet-seeker,  is  a  refractor  of  large  aperture 
and  wide  field.     Its  power  does  not  exceed  100. 


CHAPTER   III. 

ON  THE  CORRECTIONS  OF  THE  CO-ORDINATES  OF  THE  OBSERVED 
PLACE  OF  A  HEAVENLY  BODY. 

'*"•  "  •  *' 

87.  ANGLES  measured  at  the  earth's  surface  with  astronomical 
instruments,  answer  to  the  Apparent  Place  of  a  heavenly  body, 
and  are  termed  Apparent  elements.     In  astronomical  language, 
the  True  Place  of  a  heavenly  body  is  its  real  place  in  the  heavens, 
as  it  would  be  seen  from  the  centre  of  the  earth.     Angles  which 
relate  to  the  true  place  are  denominated  True  elements.     The 
apparent  co-ordinates  of  a  star  are  reduced  to  the  true,  by  the  ap- 
plication of  certain  corrections,  called  Refraction,  Parallax,  and 
Aberration. 

88.  Refraction  and  aberration  are  corrections  for  errors'  com- 

*  A  reflecting  telescope,  inferior  to  Herschel's  in  size,  the  diameter  of  the  spe- 
culum being  3  feet,  and  the  focal  length  26  feet,  but  pronounced  by  Dr.  Robinson 
superior  to  it  in  defining  power,  has,  within  a  few  years,  been  constructed  by  the 
Earl  of  Rosse,  of  Ireland.  The  same  nobleman  has  just  completed  the  construc- 
tion of  a  reflecting  telescope  of  unparalleled  dimensions,  from  the  use  of  which  im- 
portant discoveries  may  be  anticipated.  The  diameter  of  the  speculum  is  6  feet, 
and  it  has  a  focal  length  of  53  feet. 

|  See  Note  II. 

i  The  illuminating  and  space-penetrating  power  of  telescopes  may,  however,  yet 
be  greatly  increased,  and  a  greater  distinctness  and  definiteness  in  the  outline  of 
objects  may  be  obtained.  Much  may  perhaps  be  gained  also  by  setting  up  an  ob- 
servatory on  the  top  or  sides  of  some  lofty  mountain  above  the  greater  impurities 
and  disturbances  of  the  lower  regions  of  the  atmosphere,  and  under  a  tropical  sky 


44  CORRECTIONS  OF  CO-ORDIXATES. 

mitted  in  the  estimation  of  a  star's  place,  while  parallax  serves  to 
transfer  the  co-ordinates  from  the  earth's  surface  to  its  centre.  The 
object  of  the  reduction  of  observations  from  the  surface  to  the  cen- 
tre of  the  earth,  is  to  render  observations  made  at  different  places 
on  the  earth's  surface  directly  comparable  with  each  other.  Ob- 
servers occupying  different  stations  upon  the  earth  refer  the  same 
body  (unless  it  be  a  fixed  star)  to  different  points  of  the  celestial 
sphere.  Their  observations  cannot,  therefore,  be  compared  to- 
gether, unless  they  be  reduced  to  the  same  point,  and  the  centre 
of  the  earth  is  the  most  convenient  point  of  reference  that  can  be 
chosen. 

89.  The  co-ordinate  planes  or  circles  to  which  the  place  of  a 
star  is  referred,  (p.  17,)  are  not  strictly  stationary,  but,  on  the  con- 
trary, have  a  continual  slow  motion  with  respect  to  the  stars. 
Hence,  the  true  co-ordinates  of  a  star's  place  which  have  been 
found  for  any  one  epoch,  will  not  answer,  without  correction,  for 
any  other  epoch.     The  reduction  from  one  epoch  to  another  is 
effected  by  applying  two  corrections,  called  Precession  and  Nu- 
tation. 

REFRACTION. 

90.  We  learn  from  the  principles  of  Pneumatics,  as  well  as  by 
experiments  with  the  barometer,  that  the  atmosphere  gradually 
decreases  in  density  from  the  earth's  surface  upward.     We  learn 
also  from  the  same  sources,  that  it  may  be  conceived  to  be  made 
up  of  an  infinite  number  of  strata  of  decreasing  density,  concentric 
with  the  earth's  surface.     From  the  known  pressure  and  density 
of  the  atmosphere  at  the  surface  of  the  earth,  it  is  computed,  that 
by  the  laws  of  the  equilibrium  of  fluids,  if  its  density  were  through- 
out the  same  as  immediately  in  contact  with  the  earth,  its  altitude 
would  be  about  5  miles.     Certain  facts,  hereafter  to  be  mentioned, 
show  that  its  actual  altitude  is  not  far  from  50  miles.     Now,  it  is 
an  established  principle  of  Optics,  that  light  in  passing  from  a  va- 
cuum into  a  transparent  medium,  or  from  a  rarer  into  a  denser 
medium,  is   bent,  or  refracted,  towards  the  perpendicular  to  the 
surface  at  the  point  of  incidence.     It  follows,  therefore,  that  the 
light  which  comes  from  a  star,  in  passing  into  the  earth's  atmo- 
sphere, or  in  passing  from  one  stratum  of  atmosphere  into  another, 
is  refracted  towards  the  radius  drawn  from  the  centre  of  the  earth 
to  the  point  of  incidence. 

91.  Let  MmnN,  NraoO,  Oo^Q,  (Fig.  21,)  represent  successive 
strata  of  the  atmosphere.     Any  ray  Sp  will  then,  instead  of  pur- 
suing a  straight  course  Spx,  follow  the  broken  line  pabc ;  '•  being 
bent  downward  at  the  points  p,  a,  b,  c,,&c.,  where  it  enters  the 
different  strata.     But,  since  the  number  of  strata  is  infinite,  and 
the  density  increases  by  infinitely  small  degrees,  the  deflections 
apx,  bay,  &c.,  as  well  as  the  lengths  of  the  lines  pa,  ab,  &c.,  are 


KEFRACTION. 


45 


Fig.  21. 


infinitely  small ;  and 
therefore  pabc,  the 
path  of  the  ray,  is  a 
broken  line  of  an  in- 
finite number  of  parts, 
or  a  curved  line  con- 
cave towards  the 
earth's  surface,  as  it 
is  represented  in  Fig. 
22.  Moreover,  it  lies 
in  the  vertical  plane 
containing  the  origi- 
nal direction  of  the 
ray ;  for,  this  plane  is 
perpendicular  to  all 
the  strata  of  the  atmosphere,  and  therefore  the  ray  will  continue 
in  it  in  passing  from  one  to  the  other. 

92.  The  line  OS'  (Fig.  22)  drawn  tangent  to  paO,  the  curvilinear 
path  of  the  light,  at  its  lowest  point,  will  represent  the  direction  in 
which  the  light  enters  the  eye,  and  therefore  the  apparent  line  of 

Fig.  22. 


direction  of  the  star.  If,  then,  OS  be  the  true  direction  of  the  star, 
the  angle  SOS'  will  be  the  displacement  of  the  star  produced  by 
Atmospherical  Refraction.  This  angle  is  called  the  Astronomical 
Refraction,  or  simply  the  Refraction. 

Sir  ce  paO  is  concave  towards  the  earth,  OS'  will  lie  above  OS ; 
consequently,  refraction  makes  the  apparent  altitude  of  a  star 
greater  than  its  true  altitude,  and  the  apparent  zenith  distance  of 
a  star  less  than  its  true  zenith  distance.  (We  here  speak  of  the 
true  altitude  and  true  zenith  distance,  as  estimated  from  the  station 
of  the  observer  upon  the  earth's  surface.)  Thus,  to  obtain  the 
true  altitude  from  the  apparent,  we  must  subtract  the  refraction ; 
.and  to  obtain  the  true  zenith  distance  from  the  apparent,  we  must 
add  the  refraction.  As  refraction  takes  effect  wholly  in  a  vertical 
plane,  (91,)  it  does  not  alter  the  azimuth  of  a  star. 


46 


CORRECTIONS    OF    CO-ORDINATES. 


Fig.  23. 


s' 


93.  The  amount  of  the  refraction  varies  with  the  apparent  ze- 
nith distance.     In  the  zenith  it  is  zero,  since  the  light  passes  per- 
pendicularly through  all  the  strata  of  the  atmosphere :  and  it  is 
the  greater,  the  greater  is  the  zenith  distance  ;  for,  the  greater  the 
zenith  distance  of  a  star,  the  more  obliquely  does  the  light  which 
comes  from  it  to  the  eye  penetrate  the  earth's  atmosphere,  and  en- 
ter its  different  strata,  and  therefore,  according  to  a  well-known 
principle  of  optics,  the  greater  is  the  refraction. 

94.  To  find  the  amount  of  the  refraction  for  a  given  zenith  dis 
tance  or  altitude.     Let  us  first  show  a  method  of  resolving  this 
problem  by  the  general  theory  of  refraction.     According  to  this 
theory,  the  amount  of  the  refraction,  except  so  far  as  the  convexity 
of  the  strata  of  the  atmosphere  may  have  an  effect,  depends  whol- 
ly upon  the  absolute  density  of  the  air  immediately  in  contact  with 
tne  earth,  and  not  at  all  upon  the  law  of  variation  of  the  density  of 
the  different  strata ;  that  is,  the  actual  refraction  is  the  same  that 
would  take  place  if  the  light  passed  from  a  vacuum  immediately 

into  a  stratum  of  air  of  the 
density  which  obtains  at  the 
earth's  surface.  Let  us  sup  • 
pose,  then,  that  the  whole 
atmosphere  is  brought  to  the 
same  density  as  that  portion 
of  it  which  is  in  contact  with 
the  earth,  and  let  bah  (Fig. 
23)  represent  its  surface ; 
also  let  O  represent  the  sta- 
tion of  the  observer  upon  the 
earth's  surface,  and  Sa  a  ray 
incident  upon  the  atmosphere 
at  a.  Denote  the  angle  of 
refraction  O«C  by  p,  and  the 
refraction  Oax  by  r.  The 
angle  of  incidence 
Z'aS  =  Z'aS'  +  S'aS  =  OaC 

+  Oax  =p  +  r. 
Now  if  we  represent  the  in- 
dex of  refraction  of  the  atmosphere  by  m,  we  have,  by  the  laws  of 
refraction, 

sin  Z'aS  =  m  sin  OaC,  or  sin  ( p  +  r)  =  m  sin  p  ; 
developing  (App.  For.  15,) 

sin  p  cos  r  +  cos  p  sin  r  =  m  sin  p ; 
or,  dividing  by  sin  p, 

cos  r  +  cot  p  sin  r  =  m. 

But,  as  r  is  small,  we  may  take  cos  r  =  1,  and  sin  r  =  r  —  r"  sin 
1".  (App.  47.) 

Whence,  1+cotp.r"  sinl"=m,  or  r"—  _.J" '     x— —  =Atang  p; 


sin 


REFRACTION. 


47 


putting  A  = 


Let  ZCa  =  C  ;   and  ZOa  =  Z.     OaC  = 


sm  1" 

ZOa  —  ZCa,  orp  =  Z  —  C.  Substituting,  we  have  r"  =  A  tang 
(Z  —  C  ;)  or,  omitting  the  double  accent,  and  considering  r  as 
expressed  in  seconds, 

r  =  A  tang  (Z  —  C) (2) 

When  the  zenith  distance  is  not  great,  C  is  very  small  with  respect 
to  Z.  If  we  neglect  it,  we  have 

r  =  A  tang  Z (3) ; 

which  is  the  expression  for  the  refraction,  answering  to  the  suppo- 
sition that  the  surface  of  the  earth  is  a  plane,  and  that  the  light  is 
transmitted  through  a  stratum  of  uniformly  dense  air,  parallel  to 
its  surface.  We  perceive,  therefore,  that  the  refraction,  except  in 
the  vicinity  of  the  horizon,  varies  nearly  as  the  tangent  of  the  ap- 
varent  zenith  distance. 

95.  It  has  been  ascertained  by  experiment  that  m,  the  index  of 
refraction,  (the  barometer  being  =  29.6  inches,  and  the  thermome- 
ter =  50°)  =  1.0002803.     Substituting  in  equation  (3),  after  hav- 
ing restored  the  value  of  A,  and  reducing,  there  results 

,    r  =  57". 8  tang  Z (4). 

96.  With  the  aid  of  this  formula,  or  of  others  purely  theoretical, 
astronomers  have  sought  to  determine  the  precise  amount  of  the 
refraction  at  various  zenith  distances  from  observation,  and  by  col- 
lating the  results  of  their  observations  to  obtain  empirical  formulae 
that  are  more  exact. 

97.  One  of  the  simplest  methods  of  accomplishing  this  is  the  following :  When 
the  latitude  or  co-latitude  of  a  place,  and  the  polar  distance  of  a  star  which  passes 
the  meridian  near  the  zenith,  have  been  determined,  the  refraction  may  be  found 


Fig.  24. 


for  all  altitudes  from  observation  simply. 
For,  let  P  (Fig.  24)  be  the  elevated  pole, 
Z  the  zenith,  PZE  the  meridian,  HOR 
the  horizon,  S  the  true  place  of  a  star, 
and  S'  its  apparent  place.  Suppose  the 
apparent  zenith  distance  ZS'  to  have 
been  measured.  Now,  in  the  triangle 
ZPS,  ZP  the  co-latitude  and  PS  the 
polar  distance  are  known  by  hypothe- 
sis, and  the  angle  P  is  the  sidereal  time 
which  has  elapsed  since  the  star's  last 
meridian  transit,  (or,  if  the  star  be  to 
the  east  of  the  meridian,  the  difference 
between  this  interval  and  24  sidereal 
hours,)  converted  into  degrees  by  allow- 
ing 15°  to  the  hour.  Therefore  we  may 
compute  the  true  zenith  distance  ZS, 
and  subtracting  from  it  the  apparent 
zenith  distance  ZS',  we  shall  have 
the  refraction.  For  the  solution  of  this  problem  the  polar  distance  may  be 
found  by  taking  the  complement  of  the  declination  computed  from  an  ob- 
served meridian  zenith  distance,  (68 ;)  and,  since  the  upper  and  lower  transits 
of  a  circumpolar  star  take  place  at  equal  distances  from  the  pole,  the  co-lati- 
tude may  be  found  by  taking  the  half  sum  of  the  greatest  and  least  zenith  dis- 
tances of  the  pole  star.  But  it  is  obvious  that  neither  of  these  quantities  can  be 
accurately  determined,  unless  the  measured  zenith  distances  be  corrected  for  re- 


48  CORRECTIONS    OF    CO-ORDINATES 

fraction.  When,  however,  the  zenith  distances  in  question  differ  considerably  fron* 
90°,  the  corresponding  refractions  may  be  at  first  ascertained  with  considerable 
accuracy  by  means  of  equation  (4.)  When  more  correct  formulae  have  been  ob- 
tained by  this  or  any  other  process,  the  latitude  and  polar  distance,  and  therefore 
the  refraction  answering  to  the  measured  zenith  distance,  will  become  more  accu- 
rately known. 

98.  The  various  formulae  of  refraction  having  been  tested  by 
numerous  observations,  it  is  found  that  they  are  all  (though  in  dif- 
ferent degrees)  liable  to  material  errors,  when  the  zenith  distance 
exceeds  80°,  or  thereabouts.     At  greater  zenith  distances  than  this 
the  refraction  is  irregular,  or  is  frequently  different  in  amount 
when  the  circumstances  upon  which  it  is  supposed  to  depend  are 
the  same. 

99.  The  refractive  power  of  the  air  varies  with  its  density,  and 
hence  the  refraction  must  vary  with  the  height  of  the  barometer 
and  thermometer. 

100.  The  refractions  which  have  place  when  the  barometer 
stands  at  29.6  inches,  (or,  according  to  some  astronomers,  30  inch- 
es,) and  the  thermometer  at  50°,  are  called  mean  refractions. 

The  refractions  corresponding  to  any  other  height  of  the  barom- 
eter or  thermometer,  are  obtained  by  seeking  the  requisite  correc- 
tions to  be  applied  to  the  mean  refractions,  on  the  hypothesis  that 
the  refraction  is  directly  proportional  to  the  density  of  the  atmo- 
sphere. 

101.  To  save  astronomical  observers  and  computers  the  trouble 
of  calculating  the  refraction  whenever  it  is  needed,  the  mean  re- 
fractions corresponding  to  various  zenith  distances  or  altitudes  are 
computed  from  the  formulae,  as  also  the  correction's  for  the  barom- 
eter and  thermometer,  and  inserted  in  a  table.     Table  VIII  is  Dr. 
Young's  table  of  mean  refractions,  and  Table  IX  his  table  of  cor- 
rections.    The  refraction  answering  to  any  zenith  distance  not 
inserted  in  the  table  can  be  found  by  simple  proportion.     (See 
Prob.  VII.)* 

102.  On  inspecting  Table  VIII,  it  will  be  seen  that  the  refrac- 
tion amounts  to  about  34'  when  a  body  is  in  the  apparent  horizon, 
and  to  about  68"  when  it  has  an  altitude  of  45°. 

OTHER  EFFECTS  OF  ATMOSPHERICAL  REFRACTION. 

103.  Atmospherical  refraction  makes  the  apparent  distance  of 
any  two  heavenly  bodies  less  than  the  true  ;  for  it  elevates  them 
in  vertical  circles  which  continually  approach  each  other  from  the 
horizon  till  they  meet  in  the  zenith. 

104.  Refraction  also  makes  the  discs  of  the  sun  and  moon  ap- 
pear of  an  elliptical  form  when  near  the  horizon.     As  it  increases 
with  an  increase  of  zenith  distance,  the  lower  limb  of  the  sun  or 

*  The  tables  referred  to  in  the  text  may  be  found  near  the  end  of  the  book.  Th« 
problems  referred  to  are  in  Part  IV. 


PARALLAX.  49 

moon  is  more  refracted  than  the  upper,  and  thus  the  vertical  diam- 
eter is  shortened,  while  the  horizontal  diameter  remains  the  same, 
or  very  nearly  so.  This  effect  is  most  observable  near  the  hori- 
zon, for  the  reason  that  the  increase  of  the  refraction  is  there  the 
most  rapid.  The  difference  between  the  vertical  and  horizontal 
diameters  may  amount  to  i  part  of  the  whole  diameter. 

105.  When  a  star  appears  to  be  in  the  horizon,  it  is  actually  34' 
below  it,  (102  :)  refraction,  then,  retards  the  setting  and  accele- 
rates the  rising  of  the  heavenly  bodies. 

Having  this  effect  upon  the  rising  and  setting  of  the  sun,  it  must 
increase  the  length  of  the  day. 

106.  The  apparent  diameter  of  the  sun  is  about  32'  ;  as  this  is 
less  than  the  refraction  in  the  horizon,  it  follows,  that  when  the 
sun  appears  to  touch  the  horizon  it  is  actually  entirely  below  it. 
The  same  is  true  of  the  moon,  as  its  apparent  diameter  is  nearly 
the  same  with  that  of  the  sun. 

PARALLAX. 

;  107.  The  correction  for  atmospherical  refraction  having  been 
applied,  the  zenith  distance  of  a  body  is  reduced  from  the  surface 
of  the  earth  to  its  centre,  by  means  of  a  correction  called  Parallax. 

108.  Parallax  is,  in  its  most  general  sense,  the  angle  made  by 
the  lines  of  direction,  or  the  arc  of  the  celes-  Fig.  25. 

tial  sphere  comprised  between  the  places  of 
an  object,  as  viewed  from  two  different  sta- 
tions. It  may  also  be  defined  to  be  the  an- 
gle subtended  at  an  object  by  a  line  joining 
two  different  places  of  observation.  Let  S 
(Fig.  25)  represent  a  celestial  object,  and 
A,  B  two  places  from  which  it  is  viewed. 
At  A  it  will  be  referred  to  the  point  s  of  the 
celestial  sphere,  and  at  B  to  the  point  s' ; 
the  angle  BSA,  or  the  arc  ss',  is  the  paral- 
lax. The  arc  ss'  is  taken  as  the  measure 
of  the  angle  BSA,  on  the  principle  that  the 
celestial  sphere  is  a  sphere  of  an  indefinitely 
great  radius,  so  that  the  point  S  is  not  sen- 
sibly removed  from  its  centre. 

109.  The  term  parallax  is,  however,  generally  used  in  astrono- 
my in  a  limited  sense  only,  namely,  to  denote  the  angle  included 
between  the  lines  of  direction  of  a  heavenly  body,  as  seen  from  a 
point  on  the  earth's  surface  and  from  its  centre  ;  or  the  angle  sub- 
tended at  a  heavenly  body  by  a  radius  of  the  earth.     If  C  (Fig. 
26)  is  the  centre  of  the  earth,  O  a  point  on  its  surface,  and  S  a 
heavenly  body,  OSC  is  the  parallax  of  the  body. 

110.  The  parallax  of  a  heavenly  body  above  the  horizon  is  call 
ed  Parallax  in  Altitude. 

7 


50 


CORRECTIONS  OF  CO-ORDINATES. 


The  parallax  of  a  body  at  the  time  its  apparent  altitude  is  ze- 
ro, or  when  it  is  in  the  plane  of  the  horizon  is  called  the  Horizon* 
tal  Parallax  of  the  body.  Thus,  if  the  body  S  (Fig.  26)  be  sup* 

Fig.  26. 


posed  to  cross  the  plane  of  the  horizon  at  S',  OS'C  will  be  its  hori- 
zontal parallax.     OSC  is  a  parallax  in  altitude  of  this  body. 

111.  It  is  to  be  observed,  that  the  definition  just  given  of  the  hori- 


Fig.  27. 


zontal  parallax,  answers  to 
the  supposition  that  the 
earth  is  of  a  spherical  form. 
In  point  of  fact,  the  earth 
(as  will  be  shown  in  the  se- 
quel) is  a  spheroid,  and  ac- 
cordingly the  vertical  and 
the  radius  at  any  point  of 
its  surface  are  inclined  to 
each  other ;  as  represented 
in  Fig.  27,  where  OC  is  the 
radius,  and  OC'  the  verti- 
cal. The  points  Z  and  z, 
in  which  the  vertical  and 
radius  pierce  the  celestial 
sphere,  are  called,  respec- 
tively, the  Apparent  Ze- 
nith and  the  True  Zenith. 
In  perfect  strictness,  the  horizontal  parallax  is  the  parallax  at  the 
time  zOS,  the  apparent  distance  from  tie  true  zenith,  is  90°. 
No  material  error,  however,  will  be  committed  in  supposing  the 


PARALLAX  IN  ALTITUDE,  5-] 

earth  to  be  spherical,  except  when  the  question  relates  to  the  paral- 
lax of  the  moon. 

112.  Let  the  apparent  zenith  distance  ZOS=Z,  (Fig.  26,)  the 
true  zenith  distance  ZCS  =  z,  and  the  parallax  OSC  =p.     Since 

^the  angle  ZOS  is  the  exterior  angle  of  the  triangle  OSC,  we  have 

ZOS  =  ZCS  +  OSC,  and  hence  also  ZCS  =  ZOS  —  OSC  ; 
or, 

Z=z+p,  andz  =  Z  — -p  ....  (5). 

Thus,  to  obtain  the  true  zenith  distance  from  the  apparent,  we  have 
to  subtract  the  parallax ;  and  to  obtain  the  apparent  zenith  distance 
from  the  true,  to  add  the  parallax. 

Parallax,  then,  takes  effect  wholly  in  a  vertical  plane,  like  the 
refraction,  but  in  the  inverse  manner ;  depressing  the  star,  while 
the  refraction  elevates  it.  Thus,  the  refraction  is  added  to  Z,  but 
the  parallax  is  subtracted  from  it. 

113.  To  find  an  expression  for  the  parallax  in  altitude. 

(1.)  In  terms  of  the  apparent  zenith  distance. — In  the  triangle 
SOC  (Fig.  26)  the  angle  OSC  =  parallax  in  altitude  =p,  OC  =  ra- 
dius of  the  earth  =  R,  CS  =  distance  of  the  body  S  =  D,  and  COS 
=  180°  — ZOS  =  180°  —  apparent  zenith  distance  =  180°  —  Z; 
and  we  have  by  Trigonometry  the  proportion 

sin  OSC  :  sin  COS  : :  CO  :  CS ; 
whence, 

sin  p:  sin  (180°  —  Z) : :  R  :  D; 
and 

D  sinp  =  R  sin  Z ; 
or, 

R 

ship  ==  :~  sin  Z (6). 

This  equation  shows  that  the  parallax  p  depends  for  any  given 
zenith  distance  Z  upon  the  distance  of  the  body,  and  is  less  in  pro- 
portion as  this  distance  is  greater :  also,  that  for  any  given  distance 
of  the  body  it  increases  with  an  increase  in  the  zenith  distance. 
When  Z  =  90°,  p  has  its  maximum  value,  and  then  =  horizontal 
parallax  =  H ;  and  equa.  (6)  gives 

sinH  =  g (7): 

substituting,  we  have 

sinp  =  sin  H  sin  Z  .  .  .  .  (8). 

This  last  equation  may  be  somewhat  simplified.  The  distances  of 
the  heavenly  bodies  are  so  great,  that  p  and  H  are  always  very 
small  angles  ;  even  for  the  moon,  which  is  much  the  nearest,  the 
value  of  H  does  not  at  any  time  exceed  62'.  We  may,  therefore, 
without  material  error,  replace  sinp  and  sin  H  byp  and  H.  This 
being  done,  there  results, 

p  =  HsinZ  ....  (9). 


.-':" 

52  CORRECTIONS  OF  CO-ORDINATES. 

Wherefore,  the  parallax  in  altitude  equals  the  product  of  the  hor 
izontal  parallax  by  the  sine  of  the  apparent  zenith  distance. 

If  we  take  notice  of  the  deviation  of  the  earth's  form  from  that 
of  a  sphere,  Z,  in  equation  (8),  will  represent  the  apparent  distance 
from  the  true  zenith,  (111,)  and  H  the  horizontal  parallax  as  it  is 
defined  in  Art.  111. 

(2.)  In  terms  of  the  true  zenith  distance.  —  In  the  actual  state  of  astronomy,  the 
true  co-ordinates  of  the  places  of  the  heavenly  bodies  are  generally  known,  or  may 
be  obtained  by  computation  from  the  results  of  observations  already  made,  and 
from  these  there  is  often  occasion  to  deduce  the  apparent  co-ordinates.  For  this 
purpose  there  is  required  an  expression  for  the  parallax  in  altitude  in  terms  of  the 
true  zenith  distance. 

If  we  make  Z  =  z  -\-p  (112)  in  equation  (8),  we  shall  have 

sin  p  =  sin  H  sin  (z  -j-  »),  or  sin  H  =-  -  ~ 
whence, 

and 

Dividing, 

1  -{-  sin  H  _  sin  (z  -\-p)  +  sin  p  ^ 

1  —  sinH      sm(z-+-p)  —  sinp' 


1  —  —  sip 


tang'  (45»  +  J  H)  =  -  ,  <«  APP.  For.  36,  29)  ; 

whence, 

tang  (i  z  +J>)  =  tang  $z  tang2  (45°  +  $  H)  .  .  .  (10). 

This  equation  makes  known  £  z  -\-p,  from  which  we  may  obtain  p  by  subtract- 
ing i  z. 

In  order  to  be  able  to  compute  the  parallax  in  altitude  by  means  of 
formula  (9)  or  (10),  it  is  necessary  to  know  H,  the  horizontal  parallax. 

114.   To  find  the  horizontal  parallax. 

Let  0,  O'  (Fig.  27)  represent  two  stations  upon  the  same  ter- 
restrial meridian  OEO',  and  remote  from  each  other,  Z,  Z'  their 
apparent  zeniths,  and  z,  z'  their  true  zeniths,  QCE  the  equator, 
and  S  the  body  (supposed  to  be  in  the  meridian)  the  parallax  of 
which  is  to  be  found.  Let  the  angle  OSO'  =  A,  *OS  =  Z,  s'O'S 
=  Z'  ;  also  let  CO  =  R,  CO'  =  R',  CS  =  D,  the  parallax  in  alti- 
tude OSC  =p,  and  the  parallax  in  altitude  O'SC  =p'.  Now,  by 
equation  (6),  replacing  the  sine  of  the  parallax  by  the  parallax  it- 
self, (113,) 

T?  "R'  ' 

p  =  rj  sin  Z,  and  p'  —  ~r  sin  Z'  ; 

whence 

R    .    -R'   .    -      RsinZ-fR'sinZ' 


g-  ^ 

but,  (equa.  7,^ 

HR      ,       R 
= 


HORIZONTAL  PARALLAX.  53 

Substituting  this  value  of  D,  and  deducing  the  value  of  H,  we 
have 


'    '       '  ^     >' 


___ 

R  sin  Z  +  R'  sin  Z'  R  sin  Z  +  R'  sin  Z' 
It  remains  now  to  find  an  expression  for  A  in  terms  of  measura- 
ble quantities.  Let  Os  and  O's  (Fig.  27)  be  the  directions  at  O 
and  O'  of  a  fixed  star  '  which  crosses  the  meridian  nearly  at  the 
same  time  with  the  body.  Owing  to  the  immense  distance  of  the 
star,  these  lines  will  be  sensibly  parallel  to  each  other,  (27.)  Let 
the  angle  SOs,  the  difference  between  the  meridian  zenith  dis- 
tances of  the  body  and  star,  as  observed  at  O,  be  represented  by 
d,  and  let  the  same  difference  SO's  for  the  station  O',  be  represent- 
ed by  d'.  Now, 

OSO'  =  OLO'—  SQ's  =  SOs  —  SO'*,orA  =  d  —  d1. 
If  the  body  be  seen  on  different  sides  of  the  star  by  the  two  ob- 
servers, we  shall  have 

A.=d+df. 
Substituting  in  equation  (11),  there  results, 

H=    _W^L_  (12) 

RsinZ+R'sinZ' 

If  we  regard  the  earth  as  a  sphere,  R=R',  and  dividing  by  R, 
we  have 

H_        d±d'  - 

-"'  ---- 


115.  To  find  the  parallax  by  means  of  these  formulae,  each  of 
the  two  observers  must  measure  the  meridian  zenith  distance  of 
the  body,  and  also  of  a  star  which  crosses  the  meridian  nearly  at 
the  same  time  with  the  body,  and  correct  them  for  refraction.  The 
difference  of  the  two  will  be,  respectively,  the  values  of  d  and  df  ; 
and  the  corrected  zenith  distances  of  the  body  will  be  the  values 
of  Z  and  Z',  if  formula  (13)  be  used;  if  formula  (12)  be  used, 
the  measured  zenith  distances  of  the  body  must  still  be  corrected 
for  the  reduction  of  latitude,  (p.  19,  Def.  4.) 

It-  is  not  necessary  that  the  two  stations  should  be  on  precisely 
the  same  meridian  ;  for  if  the  meridian  zenith  distance  of  the  body 
be  observed  from  day  to  day,  its  daily  variation  will  become 
known  ;  then,  knowing  also  the  difference  of  longitude  of  the  two 
places,  the  following  simple  proportion  will  give  the  change  of  ze- 
nith distance  during  the  interval  of  time  employed  by  the  body  in 
moving  from  the  meridian  of  the  most  easterly  to  that  of  the  most 
westerly  station,  viz:  as  interval  (T)  of  two  successive  transits: 
diff.  of  long.,  expressed  in  time,  •(*)  :  :  variation  of  zenith  dist. 
in  interval  T  :  its  variation  in  interval  t.  This  result,  applied 
to  the  zenith  distance  observed  at  one  of  the  stations,  will  re- 
duce it  to  what  it  would  have  been  if  the  observation  had  been  made 
tn  the  same  latitude  on  the  meridian  of  the  other  station* 

116.  The  horizontal  parallax  of  a  heavenly  body  may  be  found 


54 


CORRECTIONS  OF  CO-ORDINATES. 


by  the  foregoing  process,  to  within  1"  or  2"  of  the  truth.  No 
greater  degree  of  accuracy  is  necessary  in  the  case  of  the  moon. 
But  there  are  certain  important  uses  made  of  the  horizontal  paral- 
lax of  a  body  that  will  be  noticed  hereafter,  which  require  that  the 
parallax  of  the  sun,  and  of  the  planets,  should  be  known  with  much 
greater  precision.  The  more  accurate  methods  employed  to  deter- 
mine the  parallaxes  of  these  bodies  will  be  explained  (in  principle 
at  least)  in  subsequent  parts  of  the  wrork. 

117.  In  consequence  of  the  spheroidal  form  of  the  earth,  the  hor- 
izontal parallax  of  a  body  is  somewhat  different  at  different  places. 
Let  H  and  H'  denote  the  horizontal  parallaxes  of  the  same  body, 
and  R  and  R'  the  radii  of  the  earth  at  two  different  places.  Then, 
by  equation  (7,) 


whence, 


Thus  the  parallax  at  the  equator,  called  the  Equatorial  Paral- 
lax, is  the  greatest,  and  the  parallax  at  the  pole  the  least.  The  dif- 
ference between  the  parallaxes  of  the  moon  at  the  equator  and  at 
the  pole  may  amount  to  about  12".  For  the  other  heavenly  bodies 
the  difference  is  too  small  to  be  taken  into  account. 

118.  When  the  horizontal  parallax  has  been  found  for  any  one 
distance  and  time  from  observation,  the  horizontal  parallax  for  any 
other  distance  and  time  may  be  approximately  computed,  by  means 
of  the  principle  that  the  parallax  of  a  body  is  directly  proportional 
to  its  apparent  diameter.     The  truth  of  this  principle  appears  from 
the  fact,  that  both  the  parallax  (113)  and  the  apparent  diameter  are 
inversely  proportional  to  the  same  quantity,  viz  :  the  distance  of 
the  body  from  the  earth. 

In  the  present  condition  of  astronomical  science,  when  the  hori- 
zontal parallax  of  either  one  of  the  heavenly  bodies  is  required  for 
any  particular  time,  it  may  be  obtained  by  computation,  or  from 
tables.  It  may  also  be  taken  out  of  the  Nautical  Almanac.* 

119.  The  equatorial  horizontal  parallax  of  the  moon  varies  from 
53'  48"  to  61'  24",  according  to  the  distance  of  the  moon  from  the 
earth.    The  equatorial  parallax  of  the  moon  answering  to  the  mean 
distance,  is  57'  1". 

The  horizontal  parallax  of  the  sun  varies  slightly,  from  a  change 
of  distance.  At  the  mean  distance  it  is  8".6. 

The  horizontal  parallaxes  of  the  planets  are  comprised  within 
the  limits  31",  and  0".4. 

*  The  Nautical  Almanac  is  a  collection  of  data  to  be  used  in  nautical  and  as 
tronomical  calculations,  published  annually  in  England,  and  republished  in  New 
York.  It  may  generally  be  obtained  two  or  three  yeare  previous  to  the  year  for 
which  it  is  calculated. 


ABERRATION.  55 

The  fixed  stars  have  no  parallax.* 

120.  Parallax  in  right  ascension  and  declination,  and  in  longi- 
tude and  latitude. 

Since  the  parallax  displaces  a  body  in  its  vertical  circle,  which 
is  generally  oblique  to  the  equator  and  ecliptic,  it  will  alter 
its  right  ascension  and  declination,  as  well  as  its  longitude  and  lat- 
itude. The  difference  between  the  true  and  apparent  right  ascen- 
sion is  called  the  parallax  in  right  ascension ;  the  like  differences 
for  the  other  co-ordinates  are  called,  respectively,  parallax  in  De- 
clination^ parallax  in  longitude,  and  parallax  in  latitude. 

ABERRATION. 

121.  The  celebrated  English  astronomer,  Dr.  Bradley,  com- 
menced in  the  year  1725  a  series  of  accurate  observations,  with 
the  view  of  ascertaining  whether  the  apparent  places  of  the  fixed 
stars  were  subject  to  any  direct  alteration  in  consequence  of  the 
supposed  continual  change  of  the  earth's  position  in  space.     The 
observations  showed  that  there  had  been  in  reality,  during  the  pe- 
riod of  observation,  small  changes  in  the  apparent  places  of  each 
of  the  stars  observed,  which,  when  greatest,  amounted  to  about 
40"  ;  but  they  were  not  such  as  should  have  resulted  from  the  sup- 
posed orbitual  motion  of  the  earth.     These  phenomena  Dr.  Brad- 
ley undertook  to   examine  and  reduce  to  a  general  law.     After 
repeated  trials,  he  at  last  succeeded  in  discovering  their  true  ex- 
planation.   His  theory  is,  that  they  are  different  effects  of  one  gen- 
eral cause,  a  progressive  motion  of  light  in  conjunction  with  an 
orbitual  motion  of  the  earth. 

Fig.  28. 


A      A'    A"     0       B 

122.  Let  us  conceive  the  observer  to  be  stationed  at  the  earth's 
centre  ;  and  let  ACB  (Fig.  28)  be  a  portion  of  the  earth's  orbit,  so 
small  that  it  may  be  considered  a  right  line,  CS  the  true  direction 

*  The  practical  method  of  correcting  for  parallax  is  detailed  and  exemplified  in 
Problem  VIII. 


56  CORRECTIONS  OF  CO-ORDINATES. 

of  a  fixed  star  as  seen  from  the  point  C,  AC  the  distance  through 
which  the  earth  moves  in  some  small  portion  of  time,  and  aC  the 
distance  through  which  a  particle  of  light  moves  in  the  same  time. 
Then,  a  particle  of  light,  which,  coming  from  the  star  in  the  direc- 
tion SC,  is  at  a  at  the  same  time  that  the  earth  is  at  A,  will  arrive  at 
C  at  the  same  time  that  the  earth  does.  Suppose  that  Aa  is  the 
position  of  the  axis  or  central  line  of  a  telescope,  when  the  earth 
is  at  A,  and  that,  continuing  parallel  to  itself,  it  takes  up  by  virtue 

of  the  earth's  motion,  the  successive  positions  A'a',  A."a" 

CS'.  A  particle  of  light  which  follows  the  line  SC  in  space  will 
descend  along  this  axis  :  for  aa1  is  to  AA'  and  aa"  is  to  AA",  as 
aC  is  to  AC,  that  is,  as  the  velocity  of  light  is  to  the  velocity  of 
the  earth ;  consequently,  when  the  earth  is  at  A'  the  particle  of 
light  is  on  the  axis  at  a',  and  when  the  earth  is  at  A"  the  particle 
of  light  is  on  the  axis  at  a",  and  so  on  for  all  the  other  positions  of 
the  axis,  until  the  earth  arrives  at  C.  The  apparent  direction  of 
the  star  S,  as  far,  at  least,  as  it  depends  upon  the  cause  under  con- 
sideration- will  therefore  be  CS'. 

The  angle  SCS',  which  expresses  the  change  in  the  apparent 
place  of  a  star  S,  produced  by  the  motion  of  light  combined  with 
the  motion  of  the  spectator,  is  called  the  Aberration  of  the  star  ; 
and  the  phenomenon  of  the  change  of  the  apparent  course  of  the 
light  coming  from  a  star,  thus  produced,  is  called  Aberration  of 
Light,  or  simply  Aberration. 

123.  The  phenomenon  of  the  aberration  of  light  may  be  famil- 
iarly illustrated  by  taking  falling  drops  of  rain  instead  of  particles 
of  light,  and  a  vessel  in  motion  at  sea  instead  of  the  earth  moving 
through  space  ;  and  considering  what  direction  must  be  given  to 
a  small  tube  by  a  person  standing  upon  the  deck  of  the  vessel,  so 
as  to  permit  the  drops  falling  perpendicularly  to  pass  through  the 
tube.  It  is  plain,  that  if  the  tube  had  a  precisely  vertical  position,  its 
forward  motion  would  bring  the  back  part  of  the  tube  against  the 
drop  ;  and  that  the  only  way  to  prevent  this  is  to  incline  the  upper 
end  of  the  tube  forward,  or  draw  the  lower  end  backward,  whereby 
the  back  part  of  it  would  be  made  to  pass  through  a  greater  dis- 
tance before  it  comes  up  to  the  line  of  descent  of  the  drop.     The 
quantity  that  it  is  made  to  deviate  in  direction  from  this  line  must 
depend  upon  the  relative  velocities  of  the  falling  drop  and  moving 
tube.     To  the  observer,  unconscious  of  his  own  motion,  the  drop 
will  appear  to  fall  in  the  oblique  direction  of  the  tube. 

124.  If  through  the  point  a  (Fig.  29)  a  line  as'  be  drawn  parallel 
to  AC,  and  terminating  in  CS',  the  figure  Aas'C  will  be  a  parallel- 
ogram, and  therefore  as'  will  be  equal  to  AC.     Hence  it  appears, 
that  if  on  CS,  the  line  of  direction  of  a  star  S,  a  line  Ca  be  laid  off, 
representing  the  velocity  of  light,  and  through  a  a  line  as'  be  drawn, 
naving  the  same  direction  as  the  earth's  motion  and  equal  to  its  ve- 
locity, the  line  joining  s'  and  C  will  be  the  apparent  line  of  direc- 
tion of  the  star,  the  point  S'  its  apparent  place  in  the  heavens,  and 


ABERRATION.  57 

the  angle  aCs'  its  aberration.  We  conclude,  therefore,  that  by 
virtue  of  aberration  a  star  is  seen  in  advance  of  its  true  place,  in  the 
plane  passing  through  the  line  of  direction  of  the  star  and  the  !ine 
of  the  earth's  motion. 

Fig.  29. 


The  amount  of  the  aberration  of  a  star  is  always  very  small, 
(never  greater  than  about  20",)  because  of  the  very  great  dispropor- 
tion between  the  velocity  of  light  and  the  velocity  of  the  earth.  It 
is  very  much  exaggerated  in  Figs.  28  and  29. 

125.  The  aberration  is  the  same  when  a  star  is  viewed  with  the 
naked  eye,  as  when  it  is  seen  through  a  telescope.     For,  let  «C, 
the  velocity  of  the  light,  be  decomposed  into  two  velocities,  of 
which  one,  AC,  is  equal  and  parallel  to  the  velocity  of  the  earth  ; 
the  other  will  be  represented  by  s'C.     Now,   since  the  velocity 
AC^is  equal  and  parallel  to  the  velocity  of  the  earth,  it  will  pro- 
duce no  change  in  the  relative  position  of  a  particle  of  light  and 
the  eye,  and  therefore  the  relative  motion  of  the  light  and  the  eye 
will  be  the  same  that  it  would  be  if  the  earth  were  stationary  and 
the  light  had  orjy  the  velocity  s'C  ;  accordingly,  the  light  entering 
the  eye  just  as  it  would  do  if  it  actually  came  in  the  direction  s'C, 
and  the  eye  were  at  rest,  Cs'  will  be  the  apparent  direction  of  the 
star  from  which  it  proceeds. 

126.  If  we  regard  the  observer  as  situated  upon  the  earth's 
surface,  instead  of  being  at  its  centre,  the  aberration  resulting 
from  the  earth's  motion  of  revolution  will  be  still  the  same  :  for, 
all  points  of  the  earth  advance  at  the  same  rate  and  in  the  same 
direction  with  the  centre.     The  motion  of  rotation  will  produce  an 
aberration  proper  to  itself,  but  it  is  so  small  that  there  is  no  occa- 
sion to  take  it  into  account. 

127.  To  find  a  general  expression  for  the  aberration. — We  have 
by  Trigonometry,  (Fig.  29,) 

sin  A.aC  :  sin  CA«  :   :   CA  :  Ca  :  :  vel.  of  earth  :  vel.  of  light ; 
whence, 

f1  A 

sin  AaC  —  sin  CAez  -^-,  or,  since  AaC  =  SCS', 
(^a 

.  vel.  of  earth  . 

sm  aberr.  =  sin  CAa  — = —  (14). 

vel.  of  light 

When  CAa  is  90°,  the  aberration  has  its  maximum  value,  and 
this  has  been  found  by  observation  to  be  about  20"(20".44) ;  whence, 

8 


58  CORRECTIONS    OF    CO-ORDINATES. 

vel.  of  earth 

sin  20"  =  —. — F-r-j—  .  .  .  (15): 
vel.  of  light 

substituting,  and  taking  sin  BCa  for  sin  CA<z,  to  which  it  is  very 
nearly  equal,  we  have 

sin  aberr.  =  sin  BCa  sin  20"  .  .  .  (16). 

We  may  conclude  from  this  equation,  that  the  aberration  in- 
creases with  the  angle  BCa  made  by  the  direction  of  the  star  with 
the  direction  of  the  earth's  motion ;  that  it  is  equal  to  zero  when 
this  angle  is  zero,  and  has  its  maximum  value  of  20"  (more  accu- 
rately 20". 44)  when  this  angle  is  90°. 

128.  Let  us  now  inquire  into  the  entire  effect  of  aberration  in 
the  course  of  a  year.  Let  S  (Fig.  30)  be  the  sun  ;  E  the  earth ; 
~Efg  its  orbit ;  ZTV  that  orbit  extended  to  the  fixed  stars,  or  the 
ecliptic,  (p.  15,  Def.  17 ;)  ET  a  tangent  to  the  earth's  orbit  at  E  ; 
©  the  place  of  S  among  the  fixed  stars  or  in  the  ecliptic,  as  seen 
Fig.  30.  from  the  earth  ;  s  a  fixed  star  ; 

s/T  the  arc  of  a  great  circle  pass- 
ing through  s  and  T.  Then,  by 
what  has  preceded,  (124,)  the 
earth  moving  in  the  direction 
E/g,  the  apparent  place  of  the 
star  may  be  represented  by  s'  and 
the  aberration  by  s~Es'.  Thus, 
the  effect  of  aberration  at  any  one 
time  is  to  displace  the  star  by  a 
small  amount,  directly  towards 
the  point  T  of  the  ecliptic,  which 
is  90°  behind  the  sun.  As  the 
earth  moves,  the  position  of  the  point  T  will  vary ;  and  in  the 
course  of  a  year,  while  the  earth  describes  its  entire  orbit  in  the 
direction  E/g",  this  point  will  move  in  the  same  direction  entirely 
around  the  ecliptic.  In  this  period  of  time,  therefore,  ss'  the  small 
arc  of  aberration  will  revolve  entirely  around  s  the  true  position  of 
the  star  ;  from  which  we  conclude,  that  in  consequence  of  aberra- 
tion a  star  appears  to  describe  a  closed  curve  in  the  heavens  around 
its  true  place. 

As  the  inclination  of  the  direction  of  the  star  to  the  direction  oi 
the  earth's  motion  will  vary  during  a  revolution  of  the  earth,  the 
aberration  will  also  vary  during  this  period,  (127,)  and  hence  the 
curve  in  question  will  not  be  a  circle.  It  appears  upon  investiga- 
tion that  it  is  an  ellipse,  having  the  true  place  of  the  star  for  its 
centre,  and  of  which  the  semi-major  axis  is  constant  and  equal  to 
20  '.44,  and  the  semi-minor  axis  variable  and  expressed  by  20''.44 
sin  X,  (X  denoting  the  latitude  of  the  star.)  Each  star,  then,  de- 
scribes an  ellipse  which  is  the  more  eccentric  in  proportion  as  the 
star  is  the  nearer  to  the  ecliptic ;  for,  the  expression  for  the  minoi 
axis  shows  that  the  smaller  the  latitude  the  less  will  be  this  axis. 
For  a  star  situated  in  the  ecliptic  the  minor  axis  will  be  zero,  and 


ABERRATION.  59 

the  ellipse  will  be  reduced  to  a  right  line.  For  a  star  in  the  pole 
of  the  ecliptic  the  minor  axis  is  equal  to  the  major,  and  the  ellipse 
therefore  becomes  a  circle. 

When  the  earth  is  at  two  diametrically  opposite  points  of  its  orbit,  as  E  and  r, 
the  direction  of  its  motion,  which  is  the  same  as  that  of  the  tangent  to  the  orbit, 
will  make  equal  angles  with  the  line  of  direction  of  the  star,  but  will  be  towards 
diametrically  opposite  points  in  the  sphere  of  the  heavens,  (since  the  earth's  orbit 
is  to  be  considered  as  a  mere  point  in  the  centre  of  this  sphere,  (27.)  It  follows, 
therefore,  that  in  all  such  positions  of  the  earth  the  aberration  is  the  same,  but  in 
opposite  directions.  At  E  and  r,  where  the  angle  sET  included  between  the  line 
of  direction  of  the  star  and  that  of  the  earth's  motion  is  90°,  the  aberration  is  at 
its  maximum,  and  the  star  is  at  the  extremities  of  the  major  axis  of  its  elliptic  orbit. 
At  /  and  g,  90°  distant  from  E  and  r,  this  angle  is  at  its  minimum  ;  the  aberration 
is  the  least  possible,  and  the  star  is  at  the  extremities  of  the  minor  axis  of  its  orbit. 

129.  Since  aberration  causes  the  apparent  place  of  a  star  to  dif- 
fer slightly  from  its  true  place,  the  true  and  apparent  co-ordinates 
will,  in  consequence,  differ  somewhat  from  each  other.  The  effects 
of  the  aberration  of  light  upon  the  apparent  right  ascension  and 
declination  of  a  star,  are  called, -respectively,  the  Aberration  in 
Right  Ascension  and  the  Aberration  in  Declination.  In  like  man- 
ner its  effects  upon  the  longitude  and  latitude  are  called  the  Aber- 
ration in  Longitude  and  the  Aberration  in  Latitude* 

130.  Since  the  motion  of  the  earth  is  at  all  times  in  a  direction  perpendicular,  or 
nearly  so,  to  the  line  followed  by  the  light  which  comes  from  the  sun  to  the  earth, 
the  aberration  of  the  sun,  which  takes  place  only  in  longitude,  is  continually  equal 
to  20".44,  (127.)  Thus,  the  sun's  apparent  place  is  always  20"  behind  its  true  place. 

131.  For  a  planet,  the  aberration  is   different  from  what  it  is  for  a  fixed  star. 
As  a  planet  changes  its  place  during  the  time  that  the  light  is  passing  from  it  to 
the  earth,  it  would,  if  the  earth  were  stationary,  appear  to  be  as  far  behind  its  true 
place  as  it  has  moved  during  this  interval.     This  aberration  due  to  the  motion  of 
the  planet,  combined  with  that  due  to  the  earth's  motion,  will  give  the  real  aberra- 
tion of  the  planet. 

132.  For  the  moon,  the  aberration  occasioned  by  its  motion  around  the  earth  is 
very  small.     The  earth's  motion  produces  no  lunar  aberration,  for  the  reason  that 
the  moon,  and  consequently  the  light  emitted  from  it,  partakes  of  this  motion. 

133.  If  the  apparent  places  of  a  star,  found  at  various  times,  be 
corrected  for  aberration,  the  same  result  for  the  true  place  of  the 
star  is  obtained.     Again,  the  deductions  of  Art.  128  agree  in  every 
particular  with  the  observed  phenomena  of  the  apparent  displace- 
ment of  the  stars,  first  discovered  by  Dr.  Bradley.     These  facts 
show  that  the  aberration  of  light  is  the  true  cause  of  these  phe- 
nomena, and  consequently,  at  the  same  time  establish  the  fact  of 
the  earth's  orbitual  motion,  as  well  as  that  of  the  progressive  mo- 
tion of  light. 

134.  It  may  be  worth  while  to  state,  that  the  first  discovery  of 
the  progressive  motion  of  light  preceded  the  detection  and  expla- 
nation by  Bradley  of  the  phenomena  of  aberration.     The  discov- 
ery was  made  by  Roemer,  a  Danish  astronomer,  in  the  year  1667, 
from  a  comparison  of  observations  upon  the  eclipses  of  Jupiter's 
satellites. 


*  For  the  practical  method  of  determining  and  applying  these  corrections, 
Probs.  XIX.,  XXL,  XXII.,  XXIII. 


60 


CORRECTIONS  OF  CO-ORDINATES. 


135.  As  to  the  actual  velocity  of:-;  light,  we  have,  by  equation 
(15,)  vel.  of  earth  :  vel.  of  light    :  sin  20"  .44  :  1  : :  1  :  10,000, 
(nearly.)'   Taking  the  velocity  of  the  earth  at  68,167  miles  per 
hour,  and  making  the  calculation  by  logarithms,  we  obtain  for  the 
velocity  of  light  191,000  (191,140)  miles  per  second.     As  deter- 
mined from  observations  upon  Jupiter's  satellites,  it  is  very  nearly 
the  same.     The  time  employed  by  light  in  coming  from  the  sun  to 
the  earth  is  8m.  18s. 

PRECESSION  AND  NUTATION. 

136.  In  the  investigations  that  follow,  we  shall  take  it  for  grant- 
ed that  it  is  possible  to  find  the  obliquity  of  the  ecliptic  and  the 
place  of  the  equinox.     Methods  of  determining  them  will  be  given 
when  we  come  to  treat  of  the  apparent  motion  of  the  sun. 

137.  By  comparing  the  longitudes  and  latitudes  of  the  same 
fixed  stars,  obtained  at  different  periods,  (69,)  it  is  found  that  their 
latitudes  continue  very  nearly  the  same,  but  that  all  their  longi- 
tudes increase  at  the  mean  rate  of  about  50"  per  year.     The  longi- 
tude of  a  star  being  the  arc  of  the  ecliptic,  intercepted  in  the  order 
of  the  signs  between  the  vernal  equinox  and  a  circle  of  latitude 
passing  through  the  star,  (p.  18,  Def.  30,)  it  follows  from  the  last 
mentioned  circumstancej  that  the  vernal  equinox  must  have  a  mo- 
tion along  the  ecliptic  in  a  direction  contrary  to  the  order  of  the 
signs,  amounting  to  about  50"  in  a  year.     As  it  has  been  found 
that  the  autumnal  equinox  is  always  at  the  distance  of  180°  from 
the  vernal,  it  must  have  the  same  motion.   This  retrograde  motion 
of  the  equinoctial  points,  is  called  the  Precession  of  the  Equinoxes. 

138.  As  the  latitude  of  a  star  is  its  distance  from  the  ecliptic, 
(p.  18,  Def.  31,)  it  follows  from  the  circumstance  of  the  latitudes 
of  all  the  stars  continuing  very  nearly  the  same,  that  the  ecliptic 
remains  fixed,  or  very  nearly  so,  with  respect  to  the  situations  of 
the  fixed  stars. 

139.  The  ecliptic  being  stationary,  it  is  plain  that  the  precession 

•p-    gj  of  the  equinoxes  must  result  from 

a  continual  slow  motion  of  the 
equator  in  one  direction.  It  ap- 
pears from  observation,  that  the 
obliquity  of  the  ecliptic,  or  the 
inclination  of  the  equator  to  the 
ecliptic,  remains,  in  the  course  of 
this  motion,  very  nearly  the  same. 
140.  Since  the  equator  is  in 
motion,  its  pole  must  change  its 
place  in  the  heavens.  Let  VLA 
(Fig.  31)  represent  the  ecliptic, 
K  its  pole,  which  is  stationary, 
P  the  position  of  the  north  pole 


PRECESSION.  61 

of  the  equator  or  of  the  heavens  at  any  given  time,  and  YEA  the 
corresponding  position  of  the  line  of  the  equinoxes :  KPL  re- 
presents the  circle  of  latitude  passing  through  P,  or  the  solsti- 
tial colure.  Now,  the  point  V  being  at  the  same  time  in  the 
ecliptic  and  equator,  it  is  90°  distant  from  the  two  points  K  and  P, 
the  poles  of  these  circles ;  therefore,  it  is  the  pole  of  the  circle 
KPL  passing  through  these  points,  and  hence  VL  =  90°.  It  fol- 
lows from  this,  that  when  the  vernal  equinox  has  retrograded  to  any 
point  V,  the  pole  of  the  equator,  originally  at  P,  will  be  found  in 
the  circle  of  latitude  KP'L'  for  which  V'L'  equals  90°  :  it  will 
also  be  at  the  distance  KP'  from  the  pole  of  the  ecliptic,  equal  to 
KP.  Whence  it  appears  that  the  pole  of  the  equator  has  a  retro- 
grade motion  in  a  small  circle  about  the  pole  of  the  ecliptic,  and 
at  a  distance  from  it  equal  to  the  obliquity  of  the  ecliptic.  As  the 
motion  of  the  equator  which  produces  the  precession  of  the  equi- 
noxes is  uniform,  the  motion  of  the  pole  must  be  uniform  also  ; 
and  as  the  pole  will  accomplish  a  revolution  in  the  same  time  with 
the  equinox,  its  rate  of  motion  must  be  the  same  as  that  of  the 
equinox,  that  is,  50"  of  its  circle  in  a  year.  The  period  of  the 
revolution  of  the  equinox  and  of  the  pole  of  the  equator  is  some- 
thing less  than  26,000  years. 

141.  It  is  an  interesting  consequence  of  this  motion  of  the  pole 
of  the  equator  and  heavens,  that  the  pole  star,  so  called,  will  not 
always  be  nearer  to  the  pole  than  any  other  star.     The  pole  is  at 
the  present  time  approaching  it,  and  it  will  continue  to  approach 
it  until  the  present  distance  of  1 1°  becomes  reduced  to  less  than 
i°,  which  will  happen  about  the  year  2100:  after  which  it  will 
begin  to  recede  from  it,  and  continue  to  recede,  until  about  the  year 
3200  another  star  will  come  to  have  the  rank  of  a  pole  star.    The 
motion  of  the  pole  still  continuing,  it  will,  in  the  lapse  of  centuries, 
pass  in  the  vicinity  of  several  pretty  distinct  stars  in  succession, 
and  in  about  1 3,000  years  will  be  within  a  few  degrees  of  the  star 
Vega,  in  the  constellation  of  the  Lyre,  the  brightest  star  in  the 
northern  hemisphere. 

The  present  pole  star  has  held  that  rank  since  the  time  of  the 
celebrated  astronomer  Hipparchus,  who  flourished  about  120  B.C. 
In  very  ancient  times  a  pretty  bright  star  in  the  constellation  of 
the  Dragon  (a  Draconis)  was  the  pole  star. 

142.  The  motion  of  the  equator  which  produces  the  precession 
of  the  equinoxes,  must  also  produce  changes  in  the  right  ascensions 
and  declinations  of  the  stars.     These  changes  will  be  different  ac- 
cording to  the  situations  of  the  stars  with  respect  to  the  equator 
and  equinoctial  points. 

143.  The  ecliptic,  although  very  nearly  stationary,  as  stated  in  Art.  138,  is  not 
strictly  so.  By  comparing  the  values  of  the  obliquity  of  the  ecliptic,  found  at  dis- 
tant periods,  it  is  ascertained  that  it  is  subject  to  a  gradual  diminution  from  century 
to  century.  A  comparison  of  the  results  of  observations  made  by  Flamstead  in 
1690,  and  by  Dr.  Maskelyne  in  1769,  gives  for  the  mean  secular  diminution  50" 


62  CORRECTIONS  OF  CO-ORDINATES. 

and  for  the  mean  annual  diminution  0".50.     A  more  accurate  determination  of  the 
mean  annual  diminution  is  0".4G. 

It  appears  from  observation,  that  there  are  minute  secular  changes  in  the  lati- 
tudes of  the  stars,  which  establish  that  the  diminution  of  the  obliquity  of  the  eclip- 
tic arises  from  a  slow  displacement  of  the  plane  of  the  ecliptic  (or  of  the  earth's 
orbit)  in  space. 

144.  If  the  ecliptic  slowly  changes  its  position  in  the  heavens,  its  pole  must  like- 
wise ;  and  since  the  obliquity  of  the  ecliptic  is  continually  diminishing,  its  pole 
must  be  gradually  approaching  the  pole  of  the  equator. 

145.  The  motion  of  the  ecliptic  alters  somewhat  the  precession  of  the  equinoxes, 
making  it  a  little  less  than  it  would  be  if  the  equator  only  was  in  motion  :  for,  let 

EL  (Fig.  32)  represent  the  position  of  the  eclip- 
tic, and  VQ  that  of  the  equator,  at  any  assumed 
date,  and  EL',  V'Q'  the  positions  of  the  same 
circles  at  some  later  date  ;  the  obliquity  L'V'Q' 
at  the  second  epoch  being  less  than  that  (LVQ) 
at  the  first  epoch  :  also  let  r  be  the  physical  point 
of  the  moveable  ecliptic,  which  at  the  first  epoch 
coincided  with  the  point  V,  and  n  the  point  an- 
swering to  V ;  and  Vr,  V'n  the  arcs  of  small 
circles  described  by  the  points  V  and  V.  in  the 
motion  of  the  arc  EVL  about  the  point  E.  Since 
wV'V  is  a  right  angle,  and  Q'V'V  an  acute  an- 
gle, the  point  n  must  fall  to  the  left  of  V",  and 
therefore  V"r  will  be  less  than  nr,  or  its  equal 
W,  by  the  small  arc  nV".  But  VV  is  the 
precession  on  the  fixed  ecliptic,  and  rV"  the  ac- 
tual precession.  We  learn  by  the  aid  of  Physi- 
cal Astronomy,  that  the  amount  of  annual  pro- 
cession would,  if  the  ecliptic  were  fixed,  be  50".35.  As  we  have  already  seen,  the 
actual  precession  on  the  moveable  ecliptic  is  50",  (more  accurately,  50".23.) 

146.  It  remains  for  us  now  to  take  notice  of  a  minute  inequal- 
ity in  the  motion  of  the  equator  and  its  pole,  which  we  have  thus 
far  overlooked.  Dr.  Bradley,  in  observing  the  polar  distance  of  a 
certain  star,  (y  Draconis,)  with  the  view  of  verifying  his  theory  of 
aberration,  discovered  that  the  observed  polar  distance  did  not  agree 
with  the  apparent  polar  distance,  as  computed  from  the  results  of 
previous  observation,  by  allowing  for  precession,  aberration,  and 
refraction ;  and  hence  inferred  the  existence  of  a  new  cause  of  vari- 
ation in  the  co-ordinates  of  a  star.  On  continuing  his  observations, 
he  found  that  the  polar  distance  alternately  increased  and  dimin- 
ished, and  that  it  returned  to  the  same  value  in  about  19  years. 
These  phenomena  led  him  to  suppose  that  the  pole,  instead  of 
moving  uniformly  in  a  circle  around  the  pole  of  the  ecliptic,  re- 
volved around  a  point  conceived  to  move  in  this  manner. 

If  the  pole  has  such  a  motion,  it  is  plain  that  (allowing  the  fact 
of  the  earth's  rotation)  it  must  result  from  a  vibratory  motion  of 
the  earth's  axis.  To  this  supposed  vibration  of  the  axis  of  the 
earth,  and  consequently  of  that  of  the  heavens,  Dr.  Bradley  gave 
the  name  of  Nutation.  The  term  Nutation  is  also  applied  to  the 
changes  of  the  co-ordinates  of  a  star's  place,  which  are  produced 
by  the  nutation  of  the  earth's  axis.  The  point  about  which  the 
pole  was  conceived  to  revolve,  is  the  mean  position  of  the  pole,  or 
the  Mean  Pole. 


NUTATION. 


63 


Dr.  Bradley  discovered,  from  his  observations,  that  the  curve  described  by  the 
pole  must  be  an  ellipse,  having  its  major  axis  in  the  solstitial  colure  ;  and  esti 
mated  the  value  of  the  major  axis  at  about  19",  and  that  of  the  minor  axis  at  about 
14".  He  also  discovered  that  a  connection  existed  between  the  position  of  the 
pole  in  its  ellipse,  and  the  position  of  the  moon  at  the  time  its  latitude  was  zero, 
(69,)  and  changing  from  south  to  north,  or  of  the  point  in  which  the  moon  crossed 
the  plane  of  the  ecliptic  in  passing  from  the  south  to  the  north  side  of  it,  called  the 


Fig.  33. 


ascending  node  of  the  moon's  orbit ;  for  he 
found  that  the  pole  retrograded  in  like  man- 
ner with  the  node ;  that  it  completed  its  revo- 
lution in  the  same  time,  namely,  in  about  19 
years ;  and  that  its  position  was  determinable 
from  the  place  of  the  node  by  a  geometrical 
construction.  Let  P  (Fig.  33)  represent  the 
mean  pole,  and  p  the  true  pole ;  pfg1  repre- 
sents the  ellipse  described  by  the  true  pole 
around  P  as  a  centre ;  gg',  lying  in  the  sol- 
etitial  colure  KPL,  being  its  major  axis,  and 
ff  its  minor  axis.  It  is  to  be  observed  that 
the  pole  P  is  not  stationary,  but  revolves  in 
the  circle  NPP',  carrying  with  it  the  ellipse 
pfg1.  It  will  be  seen  that  this  ellipse  is  very 
much  exaggerated  in  the  figure  :  a  true  de- 
lineation of  it  on  the  scale  of  the  figure  would 
be  altogether  imperceptible. 

This  theory  of  a  nutation  of  the  earth's  axis  has  been  verified  by  subsequent  ob- 
servations, and  Physical  Astronomy  has  revealed  the  cause  of  the  phenomenon. 

147.  As  the  equator  must  move  with  the  axis  of  the  earth  or  heavens,  nutation 
will  change  the  position  of  the  equinox  and  the  obliquity  of  the  ecliptic.  It  is  plain 
that  its  effect  upon  the  position  of  the  equinox  will  be  to  make  it  oscillate  periodi- 
cally and  by  equal  degrees,  from  one  side  to  the  other  of  the  position  which  corre- 
sponds to  the  mean  pole  ;  and  that  its  effect  upon  the  obliquity  of  the  ecliptic  will 
be  to  make  it  alternately  greater  and  less  than  the  obliquity  corresponding  to  the 
mean  pole.     The  position  of  the  equinox  which  corresponds  to  the  mean  pole,  is 
called  the  Mean  Equinox.     The  obliquity  corresponding  to  the  mean  pole,  is  term- 
ed the  Mean  Obliquity.     Mean  Equator  has  a  like  signification.    The  real  equinox 
and  the  real  equator  are  called,  respectively,  the    True  Equinox  and  the  True 
Equator.     The  actual  obliquity  of  the  ecliptic  is  termed  the  Apparent  Obliquity. 
Right  ascension  and  declination,  as  estimated  from  the  true  equator  and  true  equi- 
nox, are  called,  respectively,  True  Right  Ascension  and  True  Declination;  and 
longitude,  as  reckoned  from  the  true  equinox,  is  called  True  Longitude.     Right 
ascension,  declination,  and  longitude,  reckoned  from  the  mean  equinox  and  mean 
equator,  are  called,  respectively,  Mean  Right  Ascension,  Mean  Declination,  and 
Mean  Longitude.     The  true  and  mean  co-ordinates  differ  by  reason  of  nutation. 
The  effect  of  nutation  upon  the  right  ascension  is  called  the  Nutation  in  Right 
Ascension ;  upon  the  declination,  Nutation  in  Declination  ;  and  upon  the  longi- 
tude, Nn'.ation  in  Longitude.  Its  effect  upon  the  obliquity  of  the  ecliptic  is  called 
Nutation  of  Obliquity.     The  distance  of  the  true  from  the  mean  equinox  in  longi- 
tude, which  is  the  same  as  the  nutation  in  longitude,  is  sometimes  termed  the 
Equation  of  the  Equinoxes  in  Longitude ;  and  the  distance  in  right  ascension,  the 
Equation  of  the  Equinoxes  in  Right  Ascension.     The  precession  of  the  mean 
equinox  is  equal  to  the  Mean  Precession  of  the  true  equinox,  which  is  50" .2. 

148.  Formulae  for  computing  the  nutation  in  right  ascension,  declination,  &c., 
at  any  given  time,  are  investigated  in  some  astronomical'works.     These  formulae 
cannot  be  used  without  a  knowledge  of  the  moon's  motions.     In  practice,  the  nu- 
tations in  right  ascension,  &c.,  are  found  by  the  aid  of  tables.     (See  Probs.  XX., 
XXIII.)     If  these  be  applied  to  the  true  co-ordinates,  the  results  will  be  the  mean 
co-ordinates.     If  the  mean  co-ordinates  be  known,  the  same  corrections  will  fur- 
nish the  true. 

149.  Physical  Astronomy  has  made  known  the  existence  of  another  nutation  of 
the  earth's  axis,  too  small  to  be  detected  by  observation.    It  is  called  Solar  Nuta- 
tion. The  nutation  discovered  by  Dr.  Bradley  is  frequently  called  Lunar  Nutation 


64 


CORRECTIONS    OF    CO-ORDINATES. 


150.  To  reduce  the  co-ordinates  of  a  star  from  one  epoch  to 
another. 

This  problem  is  resolved  by  first  converting  the  true  co-ordi- 
nates into  the  mean,  then  transferring  the  mean  co-ordinates  from 
the  one  epoch  to  the  other,  and  finally  converting  the  reduced  mean 
co-ordinates  into  the  true.  The  mode  of  performing  the  first  and 
last  mentioned  operations  has  already  been  considered,  (148.)  It 
remains  now  for  us  to  show  how  to  reduce  mean  co-ordinates  from 
one  epoch  to  another. 

(I.)  When  the  interval  of  time  between  the  epochs  comprises 
but  a  few  years. — In  this  case  the  changes,  from  precession,  of 
the  mean  right  ascension  and  declination  in  the  course  of  a  year, 
called  the  Annual  Variation  in  right  ascension  and  the  Annual 
Variation  in  declination,  are  determined,  then  multiplied  by  the 
number  of  years  in  the  interval,  and  applied  as  corrections  to  the 
given  right  ascension  and  declination. 

Fig.  34.  F°r  *ms  purpose  formula?  have  been  in- 

vestigated, in  which  the  annual  variations 
in  right  ascension  and  declination  are  ex- 
pressed in  terms  of  the  right  ascension  and 
declination  of  the  star  and  the  obliquity  of 
the  ecliptic.     Let  VLA   (Fig.  34)  be  the 
ecliptic,  K  its  pole,  PFP"  the  circle  de- 
scribed by  the  mean  pole,  P  the  mean  pole 
and  VQA  the  mean  equator  at  any  given 
time,  P'   the  mean  pole  arid  V'Q'A'   the 
mean  equator  a  year  afterwards,  and  s  a 
star.     Draw  P'r  perpendicular  to  the  decli- 
nation circle  Psa.     We  have 
an.  var.  in  dec.=sa'  —  sa  =  Ps  —  P's  =  Pr  ; 
but  since  PFr  may  be  considered  as  a  right- 
angled  plane  triangle, 
Pr  =  PF  cos  P'Pr  =  PF  sin  QPa  ....  (17). 
Regarding  KPP'  as  a  right-angled  isosceles  triangle,  we  obtain 
sin  KPF  or  1 :  sin  KF  : :  sin  PKP' :  sin  PF  ; 
whence, 

sin  PF  =  sin  PKF  sin  KF,  or  PF  =  PKF  sin  KF  (nearly)  ....  (18) : 
substituting  in  equation  (17),  there  results, 

Pr  =  PKF  gin  KF  sin  QPa. 

PKP'  =  50".2  (140) ;  KF  =  obliquity  of  the  ecliptic  =  a, ; 

QPa  =  VQ  —  Va  =  90°  —  R  (R  designating  the  right  ascension  of  the  star  «.) 
Thus,  finally, 

an.  var.  in  dec.  =  50".2  sin  w  cos  R  .  .  .  .  (19). 
Next,  we  have 

an.  var.  in  r.  asc.  =  V'a'  —  Va  =  V'a'  —  mb  =  V'm  +  ba'  .  . 
but, 

V'm  =  VV  cos  V  V'm  =  50".2  cos  w  ; 
and  since  the  right-angled  triangles  sP'r  and  sba'  are  similar, 

sin  sr  or  sin  sP1  (nearly) :  sin  P'r :  :  sin  sa' :  sin  ba'  ; 
whence, 


(20)  ; 


sin6a'=  sin 


or  baf  =  P'r 


(nearly). 


The  triangle  PP'r  gives  P'r  =  PF  sin  P'Pr  =  PF  cos  QPa  ==  PKP'  sin  KF  cog 
QPa  (equa.  18)  ;  and  sin  P's  =  cos  sa'.     Substituting,  we  obtain 

ba1  =  PKF  sin  KF  cos  QPa  ^-^',  =  PKF  sin  KF  cos  QPa  tang  sa'. 


VARIATIONS  OF  THE  CORRECTIONS.  65 

Replacing  PKF,  KF,  and  QPa  by  their  values,  as  above,  and  taking  the  declina- 
tion sa  for  sa'  and  denoting  it  by  D,  there  results, 

bal  ==  50".2  sin  w  sin  R  tang  D. 
Now,  substituting  in  equation  (20)  the  values  of  V'ra  and  ba',  we  have 

an.  var.  in  r.  asc.  =  50".2  cos  w  -f-  50" .2  sin  &>  sin  R  tang  D  .  .  .  (21). 

The  results  of  formulae  (19,  21)  are  to  be  used  with  their  algebraic  signs,  if  the 
reduction  is  from  an  earlier  to  a  later  epoch,  otherwise  with  the  contrary  signs. 
The  declination  is  always  to  be  considered  positive  if  North,  and  negative  if  South. 

V'm  =  50'  .2  cos  u,  =  50".2  cos  23  o  28'  =  46".0, 
is  the  annual  retrograde  motion  of  the  equinoctial  points  along  the  equator. 

(2.)  When  the  inteival  of  the  epochs  is  of  considerable  or  great  length. — If  the 
epochs  are  separated  by  an  interval  of  more  than  10  or  12  years,  the  foregoing  pro- 
cess will  not  answer ;  for  in  a  period  of  ten  years  the  annual  variations  will  have 
sensibly  altered.*  In  this  case  we  may  proceed  as  follows  :  Convert  the  right  as- 
cension and  declination  into  longitude  and  latitude,  add  to  the  longitude  (or  if  the 
reduction  be  to  an  earlier  epoch,  subtract  from  it)  the  precession  in  longitude, 
which  will  be  the  product  of  50".23  by  the  interval  of  the  epochs,  expressed  in  years 
and  parts  of  a  year,  and  then  with  the  longitude  thus  obtained,  and  the  latitude, 
calculate  the  right  ascension  and  declination,  using  the  mean  obliquity  of  the  ecliptic. 

When  the  period  is  of  great  length,  or  very  great  precision  is  desired,  the  pre- 
cession on  the  fixed  ecliptic  should  be  used,  which  is  50".35  per  year,  (145) ;  anc 
the  right  ascension  should  be  corrected  for  the  change  of  the  position  of  the  equi- 
nox on  the  equator,  produced  by  the  motion  of  the  ecliptic ,  which  correction  is 
—  0".1313  (per  year)  for  later  epochs. 

REMARKS  ON  THE  CORRECTIONS— VERIFICATION  OF  THE 
HYPOTHESIS  THAT  THE  DIURNAL  MOTION  OF  THE  STARS 
IS  UNIFORM  AND  CIRCULAR. 

151.  It  appears  from  what  we  have  stated  on  the  subject  of  the 
Corrections  :  1 .  That  Refraction  varies  during  the  day  with  the  alti- 
tude of  the  body,  and  changes  for  all  altitudes  with  the  state  of  the 
atmosphere  ;  2.  That  Parallax  varies,  like  the  refraction,  with  the 
altitude  of  the  body,  and  changes  from  one  day  to  another  with  its 
distance ;  3.  That  Aberration  remains  sensibly  the  same  for  two 
or  three  days,  and  depends  for  its  absolute  value  on  the  time  of  the 
year ;  4.  That  Precession  and  Nutation  do  not  perceptibly  alter 
the  co-ordinates  of  a  star,  unless  it  be  a  circumpolar  star,  under 
several  days,  and  that  the  former  increases  uniformly  with  the  time 
while  the  latter  varies  periodically,  its  effects  entirely  disappearing 
in  about  19  years  ;  and,  5.  That  the  absolute  value  of  the  Nutation 
depends  entirely  upon  the  longitude  of  the  moon's  ascending  node. 

152.  In  the  determination  of  the  amount  and  laws  of  the  cor- 
rections, it  was  taken  for  granted  by  astronomers,  that  the  diurnal 
motion  of  the  stars  was  uniform  and  circular.     This  hypothesis 
may  be  verified  in  the  following  manner  :  Let  the  zenith  distance 
and  azimuth  of  the  same  star  be  measured  at  various  times  during 
a  revolution,  and  corrected  for  refraction,  (the  other  corrections  be 
ing  insensible,  (151.)  )      Then,  if  the  latitude  of  the  place  be 
known  (68)  in  the  triangle  ZPS,  (Fig.  17,  p.  37,)  we  shall  have  ZP 

*  It  is  to  be  understood  that  we  are  here  giving  methods  of  obtaining  very  accu- 
rate results.  The  process  just  explained,  except  for  stars  near  the  pole,  will  fur 
nish  results  sufficiently  accurate  for  most  purposes,  even  when  the  interval  com- 
prises 20  years  or  more. 

9 


66  OP  THE  EARTH. 

the  co-latitude,  ZS  the  zenith  distance  of  the  star,  and  PZS  its  azi- 
muth, whence  we  may  compute  PS.  If  this  calculation  be  made 
foi  the  time  of  each  observation,  it  will  be  found  that  the  same 
value  for  PS  is  obtained  in  every  instance ;  which  proves  the  di- 
urnal motion  to  be  circular.  Again,  let  the  angle  ZPS  be  com- 
puted for  the  time  of  each  observation,  with  the  same  data,  and  it 
will  be  found  that  it  varies  proportionally  to  the  time  ;  which  es- 
tablishes that  the  diurnal  motion  is  also  uniform,  or,  at  least,  sensi- 
bly so  during  one  revolution. 

153.  When  the  transits  of  a  circumpolar  star  are  observed  at 
intervals  of  several  days,  and  allowance  is  made  for  the  error  of 
the  rate  of  the  clock,  as  determined  from  observations  upon  stars 
in  the  vicinity  of  the  equator,  and  for  the  aberration  in  right  ascen- 
sion, it  is  found  that  the  sidereal  times  of  the  transits  differ  slightly 
from  each  other ;  from  which  it  appears  that  the  diurnal  motion  of 
the  stars  is  not  strictly  uniform.  When,  however,  allowance  is 
made  for  the  precession  and  nutation  in  right  ascension,  this  dif- 
ference disappears.  We  may  hence  conclude  that  the  motion  of 
rotation  of  the  earth  is  uniform,  and  that  the  motions  of  the  earth 
and  of  its  axis,  which  produce  the  phenomena  of  precession  and 
nutation,  alter  the  times  of  the  transits  of  the  stars,  thereby  making 
the  period  of  the  apparent  revolution  of  a  star  to  differ  slightly 
from  the  period  of  the  earth's  rotation. 

It  may  be  observed,  that  the  greatest  difference  obtains  in  the 
case  of  the  pole  star,  and  is  half  a  second. 


CHAPTER   IV. 

OF  THE  EARTH  ; ITS  FIGURE  AND  DIMENSIONS  I— LATITUDE  AND 

LONGITUDE  OF  A  PLACE. 

154.  ALTHOUGH  it  is  in  general  sufficient  for  astronomical  pur- 
poses to  regard  the  earth  as  a  sphere,  still  it  is  necessary  in  some 
cases  of  astronomical  observation  and  computation,  when  accurate 
results  are  desired,  to  take  notice  of  its  deviation  from  the  spheri- 
cal form.     No  account  need,  however,  be  taken  of  the  irregulari- 
ties of  its  surface,  occasioned  by  mountains  and  valleys,  as  they 
are  exceedingly  minute  when  compared  with  the  whole  extent  of 
the  earth.  It  is  to  be  understood,  then,  that  by  the  figure  of  the  earth 
is  meant  the  general  form  of  its  surface,  supposing  it  to  be  smooth, 
or  that  the  surface  of  the  land  corresponded  with  that  of  the  sea. 

155.  The  figure  of  the  earth  is  ascertained  from  an  examination 
of  the  form  of  the  terrestrial  meridians. 

A  Degree  of  a  terrestrial  meridian  is  an  arc  of  it  corresponding 
to  an  inclination  of  1°  of  the  verticals  at  the  extremities  of  the  arc. 


FIGURE  AND  DIMENSIONS  OP  THE  EARTH. 


67 


Fig.  35. 


It  is  also  called  a  Degree  of  Lat- 
itude. Thus  if  QNE  (Fig.  35) 
represent  a  terrestrial  meridian, 
ab  will  be  a  degree  of  it  if  it  be  of 
such  length  that  the  angle  aCb 
between  the  verticals  Z'«C,  Z6C, 
is  1°. 

156.  The  length  of  a  degree 
at  any  place  will  serve  as  a  meas- 
ure of  the  curvature  of  the  me- 
ridian at  that  place  ;  for  it  is  ob- 
vious, from  considerations  already 
presented,  (4,)  that  the  earth,  if 
not   strictly  spherical,   must  be 
nearly  so,   and  therefore  that  a 
degree  ab  (Fig.  35)  may,  with 

but  little  if  any  error,  be  considered  as  an  arc  of  1°  of  a  circle 
which  has  its  centre  at  C,  the  point  of  intersection  of  the  verticals 
Ca,  C6,  at  the  extremities  of  the  arc.  The  ( curvature  will  then 
decrease  in  the  same  proportion  as  the  radius  of  this  circle  in- 
creases, and  therefore  in  the  same  proportion  as  the  length  of  a 
degree  increases.  Wherefore,  the  form  of  a  meridian  may  be  de- 
termined by  measuring  the  length  of  a  degree  at  various  latitudes. 

157.  To  determine  the  length  of  a  degree  of  a  terrestrial  me- 
ridian.— To  accomplish  this,  we  have, 

(1.)  To  run  a  meridian  line ;  an  operation  which  is  performed 
in  the  following  manner.  An  altitude  and  azimuth  instrument  (or 
some  other  instrument  adapted  to  meridian  observations)  is  first 
placed  at  the  point  of  departure,  and  accurately  adjusted  to  the 
meridian.  A  new  station  is  then  established  by  sighting  forward 
with  the  telescope.  To  this  station  the  instrument  is  removed, 
and  is  there  adjusted  to  the  meridian  by  sighting  back  to  the  first 
station.  A  third  station  is  then  established  by  sighting  forward 
with  the  telescope  as  before,  to  which  the  instrument  is  removed. 
By  thus  continually  establishing  new  stations,  and  carrying  the 
instrument  forward,  the  meridian  line  may  be  marked  out  for  any 
required  distance.  The  meridian  adjustments  may  be  corrected 
from  time  to  time  by  astronomical  observations,  (51,  71.) 

(2.)  To  find  the  length  of  the  arc  passed  over. — When  the 
ground  is  level,  the  length  of  the  arc  may  be  directly  measured. 
In  case  the  nature  of  the  ground  is  such  as  not  to  allow  of  a  di- 
rect measurement,  it  may  be  calculated  with  equal  precision,  by 
means  of  a  base  line  and  a  chain  of  triangles  the  angles  of  which 
are  measured, 

(3.)  To  find  the  inclination  of  the  verticals  at  the  extreme  sta- 
tions.— This  angle  may  be  obtained  by  measuring  the  meridian 
zenith  distances  of  the  same  fixed  star  at  the  two  stations,  correct- 
ing them  for  refraction  if  they  are  observed  about  the  same  time, 


68  OF   THE    EARTH. 

and  for  refraction,  aberration,  precession,  and  nutation,  if  they  are 
observed  at  different  times,  and  taking  their  difference.     For,  let 
O,  0'  (Fig.  35)  be  the  two  stations  in  question,  Z,  Z'  their  zeniths, 
and  OS,  O'S  the  directions  of  a  fixed  star,  and  we  shall  have 
OcO'  =  ZOI  —  OIc  =  ZOS  —  Z'lS  =  ZOS  —  Z'O'S  ; 
that  is,  the  angle  comprised  between  the  verticals  equal  to  the  dif- 
ference of  the  meridian  zenith  distances  of  the  same  star. 

(4.)  The  length  of  an  arc  of  the  meridian,  either  somewhat 
greater  or  less  than  a  degree,  having  been  found  by  the  foregoing 
operations,  thence  to  compute  the  length  of  a  degree. — Let  N  de- 
note the  number  of  degrees  and  parts  of  a  degree  in  the  measured 
arc,  A  its  length,  and  x  the  length  of,  a  degree.  Then,  allowing 
that  the  earth  for  an  extent  of  several  degrees  does  not  differ  sen- 
sibly from  a  sphere,  we  may  state  the  proportion 

1°  x  A 
N  :  A  :  :  1°  :  x ;  whence  x  —  — === —  .  .  .  (22). 

158.  Degrees  have  been  measured  with  the  greatest  possible 
care,  at  various  latitudes  and  on  various  meridians.     Upon  a  com- 
parison of  the  measured  degrees,  it  appears  that  the  length  of  a 
degree  increases  as  we  proceed  from  the  equator  towards  either 
pole.     It  follows,  therefore,  (156,)  that  the  curvature  of  a  meridian 
is  greatest  at  the  equator,  and  diminishes  as  we  go  towards  the 
poles  ;  and  consequently,  that  the  earth  is  flattened  at  the  poles. 

159.  The  fact  of  the  decrease  of  the  curvature  of  a  terrestrial 
meridian  from  the  equator  to  the  poles,  leads  to  the  supposition 
that  it  is  an  ellipse,  having  its  major  axis  in  the  plane  of  the  equa- 
tor and  its  minor  axis  coincident  with  the  axis  of  the  earth.     Ana- 
lytical investigations,  founded  on  the  lengths  of  a  degree  in  differ- 
ent latitudes  and  on  different  meridians,  have  established  that  a 
meridian  is,  in  fact,  very  nearly  an  ellipse,  and  that  the  earth  has 
very  nearly  the  form  of  an  oblate  spheroid.     The  same  investiga- 
tions have  also  made  known  the  dimensions  of  the  earth.     The 
amount  of  the  oblateness  at  the  poles  is  measured  by  the  ratio  of 
the  difference  of  the  equatorial  and  polar  diameters  to  the  equato- 
rial diameter,  which  is  technically  termed  the  Oblateness. 

160.  The  form  of  the  earth  has  also  been  determined  by  other 
methods,  which  cannot  here  be  explained.     All  the  results,  taken 

together,  indicate  an  oblateness  of  — — . 

o05 

The  following  are  the  dimensions  of  the  earth  in  miles  : 

Radius  at  the  equator 3962.6  miles. 

Radius  at  the  pole 3949.6       " 

Difference  of  equatorial  and  polar  radii          13.0 
Mean  radius,  or  at  45°  latitude    .     .     .     3956.1       " 

Mean  length  of  a  degree 69.05     " 

The  fourth  part  of  a  meridian      .     .     .     6214.2       " 

161.  Owing  to  the  elliptical  form  of  a  terrestrial  meridian,  the 


LATITUDE  AND  LONGITUDE  OF  A  PLACE. 


69 


radius  and  vertical  at  a  place  do  Fig.  36. 

not  coincide.  Let  ENQS  (Fig. 
36)  represent  a  terrestrial  me- 
ridian. For  any  point  O  situa- 
ted on  this  meridian,  CO  will  be 
the  radius,  and  the  normal  line 
ZON  the  vertical.  The  posi- 
tion of  the  vertical  will  always 
be  such  that  the  apparent  zenith 
Z  will  lie  between  the  true  ze- 
nith z  and  the  elevated  pole  P. 
The  inclination  of  the  radius  to 
the  vertical,  or  the  angle  CON, 
called  the  reduction  of  latitude,  is  greatest  at  the  latitude  45°,  and 
is  there  equal  to  about  11£'. 

162.  The  oblateness  of  the  earth  occasions  some  slight  modifications  in  the 
effects  of  parallax,  which  are  in  some  instances  to  be  taken  into  account  in  com- 
puting the  apparent  azimuth  and  zenith  distance  of  a  body,  from  the  known  co- 
ordinates of  its  true  place. 


DETERMINATION  OF  THE  LATITUDE  AND  LONGITUDE  OF 

A  PLACE. 

163.  The  latitude  and  longitude  of  a  place  ascertain  its  situation 
upon  the  earth's  surface,  and  are  essential  elements  in  many  astro- 
nomical investigations. 

164.  To  find  the  latitude  of  a  place. 

(1.)  By  the  zenith  distances  or  altitudes  of  a  circumpolar  star 
at  its  upper  and  lower  transits. — The  principle  of  this  method  has 
already  been  demonstrated,  (68,)  and  shown  to  be  a  particular  case 
of  a  well  known  principle  of  Fig.  37. 

arithmetical  proportions  ;  the  fol- 
lowing is  a  more  complete  proof 
of  it.  Let  Z  (Fig.  37)  represent 
the  zenith,  HOR  the  horizon,  P 
the  pole,  and  S,  S'  the  points  at 
which  the  upper  and  lower  tran- 
sits of  a  circumpolar  star  take 
place  ;  HP  will  be  equal  to  the 
latitude,  (34,)  and  ZP  will  be^equal  to  the  co-latitude.  Now, 
we  have 

HP  =  HS  +  PS,  and  HP  =  HS'  —  PS7  =  HS'  —  PS  ; 

TTS   I  TTS' 

whence,  2HP  =  HS  +  HS',  or,  HP  =  -  - ...  (23). 

2 

In  like  manner  we  obtain 


Wherefore,  let  the  altitudes  of  a  circumpolar  star  at  its  upper  and 


70  OF   THE    EARTH. 

lower  transits  be  measured  and  corrected  for  refraction,  and  their 
half  sum  will  be  the  latitude ;  or,  let  the  zenitk  distances  be  meas- 
ured, and  corrected  for  refraction,  and  their  half  sum  subtracted 
from  90°  will  be  the  latitude.  Stars  should  be  selected  that  have 
a  considerable  altitude  at  their  inferior  transit,  for,  the  greater  is 
the  altitude  the  less  is  the  uncertainty  as  to  the  amount  of  the 
refraction.  On  this  principle  the  pole  star  is  to  be  preferred  to  all 
others. 

(2.)  By  a  single  meridian  altitude  or  zenith  distance. — Let 
5,  s',  s"  (Fig.  10,  p.  20)  be  the  points  of  meridian  passage  of  three 
different  stars,  the  first  to  the  north  of  the  zenith,  the  second  be- 
tween the  zenith  and  equator,  and  the  third  to  the  south  of  the 
equator :  ZE  =  the  latitude,  and  we  have  for  the  three  stars, 

ZE  =  sE  —  Zs,  ZE  =  s'E  +  Zs',  ZE  =  Zs"  —  s"E. 
Thus,  if  the  zenith  distance  be  called  north  or  south,  according  as 
the  zenith  is  north  or  south  of  the  star  when  on  the  meridian,  in 
case  the  zenith  distance  and  decimation  are  of  the  same  name 
their  sum  will  be  equal  to  the  latitude  ;  but  if  they  are  of  different 
names  their  difference  will  be  the  latitude,  of  the  same  name  with 
the  greater. 

This  method  supposes  the  declination  of  a  body  to  be  known. 
The  declination  of  a  star  or  of  the  sun  at  any  time  is,  in  practice, 
obtained  for  the  solution  of  this  and  other  problems,  by  the  aid  of 
tables,  or  is  taken  by  inspection  from  the  English  Nautical  Alma- 
nac, or  other  similar  work.  If  the  time  of  the  meridian  transit  be 
known,  the  altitude  may  be  measured  by  a  sextant,  (79).  The  ob- 
served altitude  must  be  corrected  for  refraction,  and  also  for  paral- 
lax if  the  body  observed  is  the  sun,  or  moon,  or  either  one  of  the 
planets. 

This  method  of  finding  the  latitude  is  the  one  most  generally 
employed  at  sea,  the  sun  being  the  object  observed.  As  the  time 
of  noon  is  not  known  with  accuracy,  several  altitudes  about  the 
time  of  noon  are  taken,  and  the  meridian  altitude  is  deduced  from 
these. 

165.  The  astronomical  latitude  being  known,  the  reduced  lati- 
tude (p.  19,  Def.  4)  may  be  obtained  by  subtracting  from  it  the 
reduction  of  latitude.  For,  if  OC  (Fig.  36)  represents  the  radius, 
and  ON  the  vertical,  at  any  place  O,  and  ECQ  represents  the  ter- 
restrial equator,  ONQ  will  be  the^stronomical  latitude,  OCQ  the 
reduced  latitude,  and  CON  the  reduction  of  latitude  ;  and  we  have 

ONQ  =  OCQ  +  CON,  and  OCQ  =  ONQ  —  CON  .  .  (25). 
(For  the  practical  method  of  resolving  this  problem,  see  Prob.  XV.) 
^     166.  There  are  various  methods  of  finding  the  longitude  of  a 
place,  nearly  all  of  which  rest  upon  the  following  principle  : 

The  difference  at  any  instant  between  the  local  times,  (whether 
sidereal  or  solar,)  at  any  place  and  on  the  first  meridian,  is  the 
longitude  of  the  place,  expressed  in  time ;  and  consequently,  a/so, 


LONGITUDE  OF  A  PLACE.  71 

the  difference  between  the  local  times  at  any  two  places  is  their 
difference  of  longitude  in  time. 

The  truth  of  this  principle  is  easily  established.  In  the  first 
place,  we  remark  that  the  longitude  of  a  place  contains  the  same 
number  of  degrees  and  parts  of  a  degree  as  the  arc  of  the  celestial 
equator  comprised  between  the  meridian  of  Greenwich  and  the 
meridian  of  the  place.  Now,  it  is  Oh.  Om.  Os.  of  mean  solar  time 
or  mean  noon  at  any  place,  when  the  mean  sun  (45)  is  on  the  me- 
ridian of  that  particular  place.  Therefore,  as  the  mean  sun,  mov- 
ing in  the  equator,  recedes  from  the  meridian  towards  the  west  at 
the  rate  of  15°  per  mean  solar  hour,  when  it  is  mean  noon  at  a 
place  to  the  west  of  Greenwich,  it  will  be  as  many  hours  and  parts 
of  an  hour  past  mean  noon  at  Greenwich,  as  is  expressed  by  the 
quotient  of  the  division  of  the  arc  of  the  celestial  equator,  or  its 
equal  the  longitude,  by  15.  If  the  place  be  to  the  east,  instead  of 
to  the  west  of  Greenwich,  when  it  is  mean  noon  there  it  will  be  as 
much  before  mean  noon  at  Greenwich  as  is  expressed  by  the  lon- 
gitude of  the  place  converted  into  time,  (as  above.)  In  either  situ- 
ation of  the  place,  then,  the  principle  just  stated  will  be  true. 

It  is  plain  that  the  equality  between  the  differences  of  the  times 
and  of  the  longitudes  will  subsist  equally  if  sidereal  instead  of  so- 
lar time  be  used. 

167.   To  find  the  longitude  of  a  place. 

(1 .)  Let  two  observers,  stationed  one  at  Greenwich  and  the  other 
at  the  given  place,  note  the  times  of  the  occurrence  of  some  phe- 
nomenon which  is  seen  at  the  same  instant  at  both  places ;  the 
difference  of  the  observed  times  will  be  the  longitude  in  time. 
These  same  observations  made  at  any  two  places  will  make  known 
their  difference  of  longitude.  If  the  stations  are  not  distant  from 
each  other,  a  signal,  as  the  flashing  of  gunpowder,  or  the  firing  of 
a  rocket,  may  be  observed.  When  they  are  remote  from  each  other, 
celestial  phenomena  must  be  taken.  Eclipses  of  the  satellites  of 
Jupiter  and  of  the  moon,  are  phenomena  adapted  to  the  purpose  in 
question.  However,  as  in  these  eclipses  the  diminution  of  the 
light  of  the  body  is  not  sudden,  but  gradual,  the  longitude  cannot 
be  obtained  with  very  great  accuracy  from  observations  made  upon 
them. 

(2.)  Transport  a  chronometer  which  has  been  carefully  adjust- 
ed to  the  local  time  at  Greenwich,  to  the  place  whose  longitude  is 
sought,. and  compare  the  time  given  by  the  chronometer  with  the 
local  time  of  the  place.  In  the  same  way,  by  transporting  a  chro- 
nometer from  any  one  place  to  another,  their  difference  of  longi- 
tude may  be  obtained.  The  error  and  rate  of  the  chronometer 
must  be  determined  at  the  outset,  and  as  often  afterwards  as  cir- 
cumstances will  admit,  that  the  error  at  the  moment  of  the  obser- 
vation may  be  known  as  accurately  as  possible.  To  ensure  greater 
certainty  and  precision  in  the  knowledge  of  the  time,  three  or  four 
chronometers  are  often  taken,  instead  of  one  only. 


72  PLACES  OF  THE  FIXED  STARS. 

This  method  is  much  used  at  sea ;  the  local  time  being  obtained 
from  an  observation  upon  the  sun  or  some  other  heavenly  body,  in 
a  manner  to  be  hereafter  explained. 

(3.)  Let  the  Greenwich  time  of  the  occurrence  of  some  celestial 
phenomenon  be  computed,  and  note  the  time  of  its  occurrence  at 
the  given  place.  ^ 

Eclipses  of  the  sun  and  moon,  and  of  Jupiter's  satellites,  occul- 
tations  of  the  stars  by  the  moon,  and  the  angular  distance  of  the 
moon  from  some  one  of  the  heavenly  bodies,  are  the  phenomena 
employed.  The  Greenwich  times  of  the  beginning  and  end  of  the 
eclipses  of  Jupiter's  satellites,  are  published  for  the  solution  of 
the  problem  of  the  longitude  in  the  English  Nautical  Almanac. 
Eclipses  of  the  sun  and  occultations  of  the  stars  furnish  the  most 
exact  determinations  of  the  longitude,  but  they  cannot  be  used 
for  this  purpose  unless  the  longitude  is  already  approximately 
known. 

The  explanation,  in  detail,  of  the  method  of  lunar  distances, 
which  is  chiefly  used  at  sea,  may  be  found  in  treatises  on  Naviga- 
tion and  Nautical  Astronomy. 


CHAPTER  V. 

OF  THE  PLACES  OF  THE  FIXED  STARS. 

168.  THE  place  of  a  fixed  star  in  the  sphere  of  the  heavens  is 
found  by  ascertaining  its  true  right  ascension  and  declination,  which 
are  the  co-ordinates  of  its  place.     The  process  of  finding  the  true 
right  ascension  and  declination  of  a  heavenly  body  has  already 
been  detailed :  the  apparent  right  ascension  and  declination  are 
found  as  explained  in  Arts.  54,  68,  and  to  these  are  applied  the 
several  corrections  of  refraction,  parallax    (when  sensible,)  and 
aberration,  (92,  120,  129.) 

When  right  ascensions  and  decimations  found  at  different  times 
are  to  be  compared  together,  or  employed  in  the  same  calculations, 
as  often  becomes  necessary,  they  are  to  be  reduced  to  the  same 
epoch  by  correcting  for  precession  and  nutation,  (p.  64.) 

169.  It  is  important  to  observe,  however,  that  the  places  of  the 
fixed  stars,  as  at  present  known,  were  not  obtained  by  the  direct 
process  just  referred  to,  that  is,  by  observing  the  right  ascension 
and  declination,  and  applying  to  them  at  once  all  the  corrections 
of  which  we  have  treated.     They  were  arrived  at  by  successive 
approximations.     The  respective  corrections  were  applied  in  suc- 
cession as  they  came  to  be  discovered ;  and  more  accurate  results 
were  obtained,  as,  by  the  improvement  of  the  instruments,  the  ob 


THE  CONSTELLATIONS.  73 

servations  became  more  and  more  exact,  and  as  the  amount  of  the 
corrections  came  to  be  known  with  greater  and  greater  precision. 

170.  In  order  to  distinguish  the  fixed  stars  from  each  other,  they 
are  arranged  into  groups,  called  Constellations,  which  are  ima- 
gined to  form  the  outlines  of  figures  of  men,  animals,  or  other  ob- 
jects, from  which  they  are  named.     Thus,  one  group  is  conceived 
to  form  the  figure  of  a  Bear,  another  of  a  Lion,  a  third  of  a  Dragon, 
and  a  fourth  of  a  Lyre.     The  division  of  the  stars  into  constella- 
tions is  of  very  remote  antiquity ;  and  the  names  given  by  the  an- 
cients to  individual  constellations  are  still  retained. 

The  resemblance  of  the  figure  of  a  constellation  to  that  of  the 
animal  or  other  object  from  which  it  is  named,  is  in  most  instances 
altogether  fanciful.  Still,  the  prominent  stars  hold  certain  definite 
positions  in  the  figure  conceived  to  be  drawn  on  the  sphere  of  the 
heavens.  Thus,  the  brightest  star  in  the  constellation  Leo  is  placed 
in  the  heart  of  the  Lion,  and  hence  it  has  sometimes  been  called 
Cor  Leonis,  or  the  Lion's  Heart :  and  the  brightest  star  in  the 
constellation  Taurus  is  situdted  in  the  eye  of  the  Bull,  and  there- 
fore sometimes  called  the  Bull's  Eye;  while  that  conspicuous 
cluster  of  seven  stars  in  this  constellation,  known  by  the  name  of 
the  Pleiades,  is  placed  in  the  neck  of  the  figure.  Again,  the  line 
of  three  bright  stars  noticed  by  every  observer  of  the  heavens  in 
the  beautiful  constellation  of  Orion,  is  in  the  belt  of  the  imaginary 
figure  of  this  bold  hunter  drawn  in  the  skies.  The  three  larger 
stars  of  this  constellation  are,  respectively,  in  the  right  shoulder,  in 
the  left  shoulder,  and  in  the  left  foot. 

171.  The  constellations  are  divided,  into  three  classes  :  North- 
ern Constellations,  Southern  Constellations,  and  Constellations  of 
the  Zodiac.     Their  whole  number  is  91  :  Northern  34,  Southern 
45,  and  Zodiacal  12.     The  number  of  the  ancient  constellations 
was  but  48.     The  rest  have  been  formed  by  modern  astronomers 
from  southern  stars  not  visible  to  the  ancient  observers,  and  others 
variously  situated,  which  escaped  their  notice,  or  were  not  atten- 
tively observed. 

172.  The  zodiacal  constellations  have  the  same  names  as  the 
signs  of  the  zodiac,  (Def.  25,  p.  17) :  but  it  is  important  to  observe 
that  the  individual  signs  and  constellations  do  not  occupy  the  same 
places  in  the  heavens.    The  signs  of  the  zodiac  coincided  with  the 
zodiacal  constellations  of  the  same  name,  as  now  defined,  about  the 
year  140  B.  C.     Since  then  the  equinoctial  and  solstitial  points 
have  retrograded  nearly  one  sign :  so  that  now  the  vernal  equinox, 
or  first  point  of  the  sign  Aries,  is  near  the  beginning  of  the  constel- 
lation Pisces  ;  the  summer  solstice,  or  first  point  of  Cancer,  near 
the  beginning  of  the  constellation  Gemini ;  the  autumnal  equinox, 
or  first  point  of  Libra,  at  the  beginning  of  Virgo  ;  and  the  winter 
solstice,  or  first  point  of  Capricornus,  at  the  beginning  of  Sagittarius, 

It  follows  from  this,  that  when  the  sun  is  in  the  sign  Aries,  he 
is  in  the  constellation  Pisces,  and  when  in  the  sign  Taurus,  in  the 

10 


74  PLACES  OF  THE  FIXED  STARS. 

constellation  Aries,  &c.,  &o.  For  the  rest,  it  should  be  observed 
that  the  constellations  and  signs  of  the  zodiac  have  not  precisely 
the  same  extent. 

173.  The  stars  of  a  constellation  are  distinguished  from  each 
other  by  the  letters  of  the  Greek  alphabet,  and  in  addition  to  these, 
if  necessary,  the  Roman  letters,  and  the  numbers  1,  2,  3,  &c. ; 
the  characters,  according  to  their  order,  denoting  the  relative  mag- 
nitude of  the  stars.     Thus,  a  Arietis  designates  the  largest  star  in 
the  constellation  Aries  ;  (3  Draconis,  the  second  star  of  the  Drag- 
on, &c. 

Some  of  the  fixed  stars  have  particular  names,  as  Sirius,  Aide- 
baran,  Arcturus,  Regulus,  &c. 

174.  The  stars  are  also  divided  into  classes,  or  magnitudes,  ac- 
cording to  the  degrees  of  their  apparent  brightness.     The  largest 
or  brightest  are  said  to  be  of  the^rs^  magnitude ;  the  next  in  order 
of  brightness,  of  the  second  magnitude ;  and  so  on  to  stars  of  the 
sixth  magnitude,  which  includes  all  those  that  are  barely  percepti- 
ble to  the  naked  eye.     All  of  a  smaller  kind  are  called  telescopic 
stars,  being  invisible  without  the  assistance  of  the  telescope.    The 
classification  according  to  apparent  magnitude  is  continued  with 
the  telescopic  stars  down  to  stars  of  the  twentieth  magnitude,  (ac- 
cording to  Sir  John  Herschel,)  and  the  twelfth  according  to  Struve. 

The  following  are  all  the  stars  of  the  first  magnitude  that  occur 
in  the  heavens,  viz.  Sirius,  or  the  Dog-star,  Betelgeux,  Rig  el,  Al- 
debaran,  Capella,  Procyon,  Regulus,  Denebola,  Cor.  Hydra, 
Spica  Virginis,  Arcturus,  Antares,  Altair,  Vega,  Deneb  or  Alpha 
Cygni,  Dubhe  or  Alpha  Ursa  Majoris,  Alpherat  or  Alpha  Andro- 
meda, Fomalhaut,  Achernar,  Canopus,  Alpha  Crucis,  and  Alpha 
Centauri.  It  is  the  practice  of  Astronomers  to  mark  more  or  less 
of  these  stars  as  intermediate  between  the  first  and  the  second 
magnitude ;  and  in  some  catalogues  some  of  them  are  assigned  to 
the  second  magnitude.  All  of  these  stars,  with  the  exception  of 
,the  last  four,  come  above  the  horizon  in  all  parts  of  the  United  States. 

175.  There  are  two  principal  modes  of  representing  the  stars  ; 
the  one  by  delineating  them  on  a  globe,  where  each  star  occupies 
the  spot  in  which  it  would  appear  to  an  eye  placed  in  the  centre 
of  the  globe,  and  where  the  situations  are  reversed  when  we  look 
down  upon  them  ;  the  other  is  by  a  chart  or  map,  where  the  stars 
are  generally  so  arranged  as  to  be  represented  in  positions  similar 
to  their  natural  ones,  or  as  they  would  appear  on  the  internal  con- 
cave surface  of  the  globe.*     The  construction  of  a  globe  or  chart 
is  effected  by  means  of  the  right  ascensions  and  declinations  of  the 
stars.     Two  points  diametrically  opposite  to  each  other  on  the 
surface  of  an  artificial  globe  are  taken  to  represent  the  poles  of  the 
heavens,  and  a  circle  traced  90°  distant  from  these  for  the  equator : 
another  point  23  £°  from  one  of  the  poles  is  then  fixed  upon  for  one 

*  Encyclopedia  Metropolitana,  Art.  Astronomy,  p.  505. 


RIGHT  ASCENSION  AND  DECLINATION.  7$ 

of  the  poles  of  the  ecliptic,  and  with  this  point  as  a  geometrical 
pole  a  great  circle  described ;  the  points  of  intersection  of  the 
two  circles  will  represent  the  equinoctial  points.  The  point  which 
represents  the  place  of  a  star  is  found  by  marking  off  the  right  as- 
cension and  decimation  of  the  star  upon  the  globe. 

All  the  fixed  stars  visible  to  the  naked  eye,  together  with  some 
of  the  telescopic  stars,  are  represented  on  celestial  globes  of  1 2  or 
18  inches  in  diameter. 

176.  The  places  of  the  fixed  stars  are  generally  expressed  by 
their  right  ascensions  and  declinations,  but  sometimes  also  by 
their  longitudes  and  latitudes.  A  table  containing  a  list  of  fixed 
stars  designated  by  their  proper  characters,  and  giving  their  mean 
right  ascensions  and  declinations,  or  their  mean  longitudes  and  lati- 
tudes, is  called  a  Catalogue  of  those  stars.* 

Table  XC.  is  a  catalogue  of  fifty  principal  fixed  stars,  and  gives 
their  mean  right  ascensions  and  declinations  for  the  beginning  of 
the  year  1840,  as  well  as  their  annual  variations  in  right  ascension 
and  declination.  The  annual  variations  serve  to  extend  the  use  of 
the  catalogue  about  10  years  (150)  before  and  after  the  epoch  for 
which  it  is  constructed.  (See  Prob.  XVIII.)  Every  ten  years,  or 
thereabouts,  a  new  catalogue  must  be  formed. 

177.  If  the  true  right  ascension  and  declination  of  a  star  at  a  given  time  be  re- 
quired, correct  the  mean  right  ascension  and  declination  found  by  the  catalogue, 
for  nutation.  (See  Art.  148.)  And  if  the  apparent  right  ascension  and  declination 
be  required,  correct  also  for  aberration.  (See  Art.  129.) 

178.  The  latitude  and  longitude  of  a  fixed  star  or  other  heavenly 
body  are  obtained  originally  by  computation  from  its  right  ascen- 
sion and  declination. 

To  convert  the  right  ascension  and  declination  of  a  body  into 
its  longitude  and  latitude. — Let  EQ  (Fig.  38)  represent  the  equa- 
tor, EC  the  ecliptic,?,  K  the  poles  of 
the  equator  and  ecliptic,  E  the  vernal 
equinox,  PSR  a  circle  of  declination 
and  KSL  a  circle  of  latitude,  both 
passing  through  a  body  S.  The 
right  ascension  of  the  body  is  ER  = 
R ;  the  declination  RS  =  D  ;  the 
longitude  EL  =  L  ;  and  the  latitude 
LS  =  X.  REL  =  w  is  the  obliqui- 
ty of  the  ecliptic,  which  is  one  of 
the  essential  data  of  the  problem. 

*  Various  catalogues  have  at  different  periods  been  published.  The  first  was  be- 
gun  by  Hipparchus,  120  years  before  the  Christian  era.  Of  the  modern  catalogues, 
the  following  may  be  cited  as  among  the  most  accurate,  although  not  the  most 
extensive,  viz.  the  Catalogues  of  Flamstead,  Lacaille,  Bradley,  Maskelyue,  Piazzi, 
and  of  the  Royal  Astronomical  Society,  and  of  the  British  Association. 

The  Nautical  Almanac  contains  a  Catalogue  of  100  principal  fixed  stars,  of 
which  54.  are  designated  as  Standard  Stars — that  is,  stars  whose  places  are  sup- 
posed to  be  known  with  all  attainable  precision.  The  largest  single  catalogue  evej 
published  is  the  Histoire  Celeste  of  Lalande,  which  gives  the  places  of  50,OQO  stars 


76 


PLACES  OF  THE  FIXED  STARS. 


RES  =  x  and  LES  =  y  are  employed  as  auxiliary  angles.  In  the 
right-angled  spherical  triangle  LES  we  have  by  Napier's  rules  for 
the  solution  of  right-angled  triangles,  (see  Appendix,) 

sin  (co.  LES)  =  tang  EL  tang  (co.  ES)  ; 
whence, 

tan  EL  =  cos  LES  tan  ES,  or,  tan  L  =  cos  (RES  —  w)  tan  ES  ; 
but 


sin  (co.  RES)  =tan  ER  tan  (co.  ES,)  or,  tan  ES  = 

COS 

thus, 

T  /Tj-nci         \  tan£  ER     cos  (x  —  w)  tan  R         ,  ^ 

tan  L  =  cos  (RES  —  w)  —  -%=^  =  -  —  -  .  .  (26): 

cos  RES  cos  x 

and  to  find  a?,  we  have 

sin  ER  =  tan  (co.  RES)  tan  RS,  or,  cot  x  =  sin  R  cot  D  .  .  (27.) 

Again, 

sin  EL  =  tan  (co.  LES)  tan  LS,  or  tan  LS  =  tan  LES  sin  EL, 

which  gives 

tang  X  =  tang  (x  —  w)  sin  L  .  .  .  (28.) 

Equation  (27)  makes  known  the  value  of  x,  with  which  we  de- 
rive the  values  of  L  and  X  by  means  of  equations  (26)  and  (28.) 
In  resolving  the  equations  attention  must  be  paid  to  the  signs  of 
the  quantities,  which  are  determined  according  to  the  usual  trigo- 
nometrical rules,  it  being  understood  that  the  declination  D  is  to 
be  regarded  as  negative  when  it  is  south,  x  is  to  be  taken  always 
less  than  180°,  and  greater  or  less  than  90°  according  as  its  cotan- 
gent is  negative  or  positive.  L  will  always  be  in  the  same  quad- 
rant with  R.  The  latitude  X  will  be  north  or  south  according  as 
tang  X  comes  out  positive  or  negative. 

The  apparent  or  mean  obliquity  is  used,  according  as  the  case 
refers  to  true  or  mean  co-ordinates.  (For  exemplifications  of  this 
problem  see  Prob.  XXIV.) 

179.  It  is  now  frequently  necessary  to  resolve  the  converse  problem,  that  is,  to 
convert  the  longitude  and  latitude  of  a  body  into  its  right  ascension  and  decli- 
nation. 
The  triangle  RES  (Fig.  38)  gives 

sin  (co.  RES)  =  tang  ER  tang  (co.  ES)  ; 
whence, 

tan  ER  =  cos  RES  tan  ES,  or,  tan  R  =  cos  (LES  -j-  «)  tan  ES  ; 
but 

sin  (co.  LES)  =  tang  EL  tang  (co.  ES),  or  tan  ES  = 
thus, 


tang  R  =  eis  (LES  +  »)  *Sft  =  <*«  (y  +  ")  tang  L    _  ,  _  _    2 

'  cos  LES  cos  y 

and  to  find  y,  we  have 

sin  EL  =  tang  (co.  LES)  tang  LS,  or  cot  y  =  sin  L  cot  X  .  .  (30). 
For  the  declination,  we  have 

sin  ER  =  tan  (co.  RES)  tan  RS,  or,  tan  RS  =  tan  RES  sin  ER  ; 
«r, 

tang  D  =  tang  (y  +  w)  sin  R  .  .  .  (31.) 


OBLIQUITY    OF    THE    ECLIPTIC.  77 

The  value  of  y  being  derived  from  equation  (30)  and  substituted  in  equations 
(29)  and  (31),  these  equations  will  then  make  known  the  values  of  R  and  D.  The 
signs  of  the  quantities  are  determined  by  the  usual  trigonometrical  rules,  the  lati- 
tude A  being  taken  negative  when  south,  y  is  always  less  than  180°,  and  greater 
or  less  than  90°  according  as  its  cotangent  comes  out  negative  or  positive.  R  will 
be  in  the  same  quadrant  as  L.  The  declination  will  be  north  or  south  according 
as  its  tangent  comes  out  positive  or  negative.  (For  exemplifications  of  this  prob- 
lem see  Prob.  XXV.) 

180.  Table  XCII.  contains  the  mean  longitudes  and  latitudes 
of  some  of  the  principal  fixed  stars  for  the  beginning  of  the  year 
1840,  together  with  their  annual  variations,  which  serve  to  make 
known  the  mean  longitudes  and  latitudes  at  any  other  epoch.  (See 
Prob.  XVIII.) 

181.  The  fixed  stars,  so  called,  are  not  all  of  them,  rigorously 
speaking,  fixed  or  stationary  in  the  heavens.     It  has  been  discov- 
ered that  many  of  them  have  a  very  slow  motion  from  year  to  year. 
These  motions  of  the  stars  are  called  their  Proper  Motions.     The 
annual  variations  in  right  ascension  and  declination,  and  in  longi- 
tude and  latitude,  given  in  Tables  XC.  and  XCII.,  are  the  varia- 
tions due  both  to  the  precession  of  the  equinoxes  and  the  proper 
motions  of  the  stars. 


CHAPTER    VI. 

OF  THE  APPARENT  MOTION  OF  THE  SUN  IN  THE  HEAVENS. 

182.  THE  sun's  declination,  and  the  difference  of  right  ascension 
of  the  sun  and  some  fixed  star,  found  from  day  to  day  throughout 
a  revolution,  are  the  elements  from  which  the  circumstances  of  the 
sun's  apparent  motion  are  derived. 

The  motion  of  the  sun,  as  at  present  known,  has  been  arrived  at 
in  the  same  approximative  manner  as  the  places  of  the  fixed  stars, 
(169.)  It  would,  in  fact,  be  theoretically  impossible  to  correct  the 
co-ordinates  of  the  sun's  apparent  place  for  precession,  nutation, 
and  aberration,  in  the  original  determination  of  the  sun's  motion  ; 
for,  the  knowledge  of  these  corrections  presupposes  some  know- 
ledge of  the  motion  of  the  sun. 

183.  The  curve  on  the  sphere  of  the  heavens  passing  through 
the  successive  positions  determined  as  above  from  day  to  day,1  is 
the  ecliptic.     If  we  suppose  it  to  be  a  circle,  as  it  appears  to  be, 
its  position  will  result  from  the  position  of  the  equinoctial  points 
and  its  obliquity  to  the  equator. 

184.  To  find  the  obliquity  of  the  ecliptic. — Let  EQA  (Fig.  39) 
represent  the  equator,  EGA  the  ecliptic,  and  OC,  OQ  lines  drawn 
through  0  the  centre  of  the  earth  and  perpendicular  to  AGE  the 


78 


APPARENT  MOTION  OF  THE  SUN. 


Fig.  39. 


line  of  the  equinoxes  ;  then  the  angle  COQ  will  be  the  obliquity 
of  the  ecliptic.     This  angle  has  for  its  measure  the  arc  CQ,  and 

therefore  the  obliquity  of  the  eclip- 
tic is  equal  to  the  greatest  decli- 
nation of  the  sun.  It  can  but 
rarely  happen  that  the  time  of  the 
greatest  declination  will  coincide 
with  the  instant  of  noon  at  the 
place  where  the  observations  are 
made,  but  it  must  fall  within  at 
least  twelve  hours  of  the  noon  for 
which  the  observed  declination 
is  the  greatest.  In  this  interval 
the  change  of  declination  cannot 
exceed  4",  and  therefore  the  greatest  observed  declination  cannot 
differ  more  than  4"  from  the  obliquity.  A  formula  has  been  in- 
vestigated, which  gives  in  terms  ol  determinable  quantities  the 
difference  between  any  of  the  greater  declinations  and  the  maxi- 
mum declination.  By  reducing  by  means  of  this  formula  a  num- 
ber of  the  greater  declinations  to  the  maximum  declination,  and 
taking  the  mean  of  the  individual  results,  a  very  accurate  value  of 
the  obliquity  may  be  found. 

185.   To  find  the  position  of  the  vernal  or  autumnal  equinox. 
(1.)  On  inspecting  the  observed  declinations  of  the  sun,  it  is  seen 
that  about  the  21st  of  March  the  declination  changes  in  the  inter- 
val of  two  successive  noons  from  south  to  north.     The  vernal 


Fig.  40. 


S:RS 


equinox  occurs  at  some  moment 
of  this  interval.  Let  RS,  R'S' 
(Fig.  40)  represent  the  declinations 
at  the  noons  between  which  the 
equinox  occurs  :  as  one  is  north 
and  the  other  south,  their  sum  (S) 
will  be  the  daily  change  of  declina- 
tion at  the  time  of  the  equinox. 
Denote  the  time  from  noon  to  noon 
by  T.  Now,  to  find  the  interval 
(x)  between  the  noon  preceding 
the  equinox  and  the  instant  of  the 
equinox,  state  the  proportion 
T.  _TxRS 

1  •* s 


on  the  principle  that  the  declination  changes  for  a  day  or  more  pro- 
portionally to  the  time.  Next,  take  the  daily  change  in  right 
ascension  (RR')  on  the  day  of  the  equinox  and  compute  the  value 
of  RE,  by  the  proportion 

rp    v  -po 

T  :  x,  or  A  : :  RR' :  RE  ; 


POSITION  OP  THE  EQUINOX.  79 

Bdd  RE  to  MR,  the  observed  difference  of  right  ascension 
(182)  on  the  day  preceding  the  equinox,  and  the  sum  ME  will  be 
the  distance  of  the  equinox  from  the  meridian  of  the  star  observed 
in  connection  with  the  sun.* 

The  position  of  the  autumnal  equinox  may  be  found  by  a  simi- 
lar process,  the  only  difference  in  the  circumstances  being  that  the 
declination  changes  from  north  to  south  instead  of  from  south  to 
north. 

If  the  value  of  x  which  results  from  the  first  proportion  be  add- 
ed to  the  time  of  noon  on  the  day  preceding  the  equinox,  the  result 
will  be  the  time  of  the  equinox. 

(2.)  In  the  triangle  RES  (Fig.  39)  we  have  the  angle  RES  =  u 
the  obliquity  of  the  ecliptic,  and  RS  =  D  the  declination  of  the 
sun,  both  of  which  we  may  suppose  to  be  known,  and  we  have  by 
Napier's  first  rule, 

sin  ER  =  tang  (co.  RES)tangRS  =  cot  wtang  D  .  .  (32 ;) 
whence  we  can  find  ER.  And  by  taking  the  sum  or  difference  of 
ER  and  MR,  according  as  the  star  observed  is  on  the  opposite 
side  of  the  sun  from  the  equinox  or  the  same  side,  we  obtain  ME 
as  before.  If  this  calculation  be  effected  for  a  number  of  posi- 
tions S,  S',  S",  &c.,  of  the  sun  on  different  days,  and  a  mean  of 
all  the  individual  results  be  taken,  a  more  exact  value  of  ME  will 
be  obtained. 

ME  being  accurately  known,  the  precise  time  of  the  equinox 
may  readily  be  deduced  from  the  observed  daily  variation  of  right- 
ascension  on  the  day  of  the  equinox. 

186.  The  calculations  just  mentioned  rest  upon  the  hypothesis 
that  the  ecliptic  is  a  great  circle.     The  close  agreement  which  is 
found  to  subsist  between  the  values  of  M  E  deduced  from  obser- 
vations upon  the  sun  in  different  positions  S,  S',  S",  &c.,  estab- 
lishes the  truth  of  this  hypothesis.     It  is  also  confirmed  by  the 
fact,  that  the  right  ascensions  of  the  vernal  and  autumnal  equinox 
differ  by  1 80°,  since  we  may  infer  from  this  that  the  line  of  the 
equinoxes  passes  through  the  centre  of  the  earth. 

187.  The  mean  obliquity  of  the  ecliptic  is  derived  from  the  apparent  obliquity,  as 
well  as  the  mean  equinox  from  the  true  equinox,  by  correcting  for  nutation. 

188.  The  mean  obliquity  at  any  one  epoch  having  been  found,  its  value  at  any 
assumed  time  may  be  deduced  from  this  by  allowing  for  the  annual  diminution  of 
0".46,  (see  Table  XXII.)    In  like  manner,  the  place  of  the  mean  equinox^at  any 
given  time  may  be  derived  from  its  place  once  found,  by  allowing  for  the  annual 
precession  of  50".23. 

The  mean  obliquity  having  thus  been  found  for  any  assumed  time,  the  apparent 
obliquity  at  the  same  time  becomes  known,  by  applying  the  nutation  of  obliquity. 
(See  Prob.  X.) 

189.  The  longitude  of  the  sun  may  be  expressed  in  terms  of 
the  obliquity  of  the  ecliptic  and  the  right  ascension  or  declination 
In  the  triangle  ERS,  (Fig.  39,)  ES(=L)  represents  the  longi- 

»  The  star  is  here  supposed  to  be  to  the  west  of  the  sun. 


80  APPARENT  MOTION  OF  THE  SUN. 

tilde  of  the  sun  supposed  to  be  at  S,  ER  (=R)  its  right  ascension 
and  RS  (=  D)  its  declination.     Now,  by  Napier's  first  rule, 


thus, 

cotL==coswcotR,ortanffL  =  —  ^—  .      .  (33). 

cos  w 

Also,  (Napier's  second  rule,  Appendix,) 

sin  RS  =  cos  (co.  RES)  cos  (co.  ES);  whence,  sin  ES  = 

or, 

.    ,       sinD  ,     . 

smL=—  —  .  .  .  (34). 
sin  w 

With  these  formulae  the  longitude  of  the  sun  may  be  computed 
from  either  its  right  ascension  or  declination.  (See  Prob.  XII.) 

Formulae  (33)  and  (34)  may  be  written  thus, 

tang  R  =  tang  L  cos  w  ;  sin  D  =  sin  L  sin  w  .  .  .  (35). 

These  formulae  will  make  known  the  right  ascension  and  decli- 
nation of  the  sun,  when  his  longitude  is  given.  (See  Prob.  XL) 
It  will  be  seen  in  the  sequel  that  in  the  present  advanced  state  of 
astronomical  science,  the  longitude  of  the  sun  at  any  assumed  time 
may  be  computed  from  the  ascertained  laws  and  rate  of  the  sun's 
motion. 

190.  The  interval  between  two  successive  returns  of  the  sun  to 
the  same  equinox,  or  to  the  same  longitude,  is  called  a  Tropical 
Year. 

And  the  interval  between  two  successive  returns  of  the  sun  to 
the  same  position  with  respect  to  the  fixed  stars,  is  called  a  Side- 
real Year. 

191.  It  appears  from  observation  that  the  length  of  the  tropical 
year  is  subject  to  slight  periodical  variations.     The  period  from 
which  it  deviates  periodically  and  equally  on  both  sides,  is  called 
the  Mean  Tropical  Year.     As  the  changes  in  the  length  of  the 
true  tropical  year  are  very  minute,  the  length  of  the  mean  tropical 
year  is  obviously  very  nearly  equal  to  the  mean  length  of  the  true 
tropical  year  in  an  interval  during  which  it  passes  one  or  more 
times  through  all  its  different  values.     In  point  of  fact,  it  may  be 
found  with  a  very  close  approximation  to  the  truth  by  comparing 
two  equinoxes  observed  at  an  interval  of  60  or  100  years. 

Theory  shows  that  the  variation  in  the  length  of  the  tropical  year  arises  from 
the  periodical  inequality  in  the  precession  of  the  equinoxes  which  results  from  nu- 
tation, and  certain  periodical  inequalities  in  the  sun's  yearly  rate  of  motion  ;  and 
thus  establishes  also,  that  the  mean  tropical  year,  as  above  defined,  is  the  same  as 
the  interval  between  two  successive  returns  of  the  sun,  supposed  to  have  its  mean 
motion,  to  the  same  mean  equinox. 

According  to  the  most  accurate  determinations,  the  length  of  the 
mean  tropical  year,  expressed  in  mean  solar  time,  is  365d.  5h. 
48m.  47.58s.,  (48s.  nearly.) 


SUN'S  DAILY  MOTION  IN  LONGITUDE.  81 

192.  In  a  mean  tropical  year  the  sun's  mean  motion  in  longi- 
tude is  360° ;  hence,  to  find  his  mean  daily  motion  in  longitude 
we  have  only  to  state  the  proportion 

365d.  5h.  48m.  48s. :  Id. : :  360°  :  x  =  59'  8".33. 

193.  The  sidereal  year  is  longer  than  the  tropical. — For  since 
the  equinox  has  a  retrograde  motion  of  50". 23  in  a  year,  when  the 
sun  has  returned  to  the  equinox  it  will  not  have  accomplished  a  si- 
dereal revolution,  into  50". 23.     The  excess  of  the  sidereal  over 
the  tropical  year  results  from  the  proportion 

59'  8".3  :  50".23  :  :  Id. :  x  =  20m.  23.1s. 

Thus  the  length  of  the  mean  sidereal  year,  expressed  in  mean 
solar  time,  is  365d.  6h.  9m.  11s. 

194.  If  from  the  right  ascensions  and  declinations  of  the  sun, 
found  on  two  successive  days,  the  corresponding  longitudes  be  de- 
duced (equas.  33,  34)  and  their  difference  taken,  the  result  will  be 
the  sun's  daily  motion  in  longitude  at  the  time  of  the  observations. 
The  sun's  daily  motion  in  longitude  is  not  the  same  throughout 
the  year,  but,  on  the  contrary,  is  continually  varying.  It  gradually 
increases  during  one  half  of  a  revolution,  and  gradually  decreases 
during  the  other  half,  and  at  the  end  of  the  year  has  -recovered  its 
original  value.     Thus,  the  greatest  and  least  daily  motions  occur 
at  opposite  points  of  the  ecliptic.     They  are,  respectively,  61'  10" 
and  57'  11". 

195.  The  exact  law  of  the  sun's  unequable  motion  can  only  be 
obtained  by  taking  into  account  the  variation  of  his  distance  from 
the  earth ;  for  the  two  are  essentially  connected  by  the  physical 
law  of  gravitation,  which  determines  the  nature  of  the  earth's  mo- 
tion of  revolution  around  the  sun. 

That  the  distance  of  the  sun  from  the  earth  is  in  fact  subject  to 
a  variation,  may  be  inferred  from  the  observed  fact,  that  his  ap- 
parent diameter  varies.  On  measuring  with  the  micrometer  the 
apparent  diameter  of  the  sun  from  day  to  day  throughout  the  year, 
it  is  found  to  be  the  greatest  when  the  daily  angular  motion,  or  in 
longitude,  is  the  greatest,  and  the  least  when  the  daily  motion  is 
the  least  *  and  to  vary  gradually  between  these  two  limits.  Ac- 
cordingly, the  sun  is  nearest  to  us  when  his  daily  angular  motion 
is  the  most  rapid,  and  farthest  from  us  when  his  daily  motion  is 
the  slowest.  The  greatest  apparent  diameter  of  the  sun  is  32' 
36'  ;  and  the  least  apparent  diameter  31'  31". 

11 


MOTIONS  OF  THE  PLANETS  IN  SPACE. 


CHAPTER  VII. 

OP  THE  MOTIONS  OF  THE  SUN,  MOON,  AND  PLANETS,  IN 
THEIR  ORBITS. 

KEPLER'S  LAWS. 

196.  THE  celebrated  astronomer  Kepler,  who  flourished  early 
in  the  seventeenth  century,  by  examining  the  observations  upon 
the  planets  that  had  been  made  by  the  renowned  Danish  observer, 
Tycho  Brahe,  discovered  that  the  motions  of  these  bodies,  and  of 
the  earth,  were  in  conformity  with  the  following  laws  : 

(1.)  The  areas  described  by  the  radius-vector  of  a  planet  [or 
the  line  drawn  from  the  sun  to  the  planet]  are  proportional  to  the 
times. 

(2.)  The  orbit  of  a  planet  is  an  ellipse,  of  which  the  sun  occu- 
pies one  of  the  foci. 

(3.)  The  squares  of  the  times  of  revolution  of  the  planets  are 
proportional  to  the  cubes  of  their  mean  distances  from  the  sun,  or 
of  the  semi-major  axes  of  their  orbits. 

These  laws  are  known  by  the  denomination  of  Kepler's  Laws. 
They  were  announced  by  Kepler  as  the  fundamental  laws  of  the 
planetary  motions,  after  a  partial  examination  only  of  these  mo- 
tions. They  have  since  been  completely  verified  by  other  astron- 
omers. We  shall  adopt  the  first  two  laws  for  the  present  as  hy- 
potheses, and  show  in  the  sequel  that  they  are  verified  by  the 
results  deducible  from  them. 

These  laws  being  established,  the  third  is  obtained  by  simply 
comparing  the  known  major  axes  and  times  of  revolution. 

197.  The  apparent  motion  of  the  sun  in  space  must  be  subject 
to  Kepler's  first  two  laws ;  for  the  apparent  orbit  of  the  sun  is  of 
the  same  form  and  dimensions  as  the  actual  orbit  of  the  earth,  and 
the  law  and  rate  of  the  sun's  motion  in  its  apparent  orbit,  are  the 
same  as  the  law  and  rate  of  the  earth's  motion.     To  establish  these 

Fig.  41.  two  facts,  let  EE'A    (Fig. 

41)  represent  the  elliptic  or- 
bit of  the  earth,  and  S  the 
position  of  the  sun  in  space. 
If  the  earth  move  from  E  to 
,0  any  point  E',  as  it  seems  to 
remain  stationary  at  E,  it  is 
plain  that  the  sun  will  ap- 
pear to  move  from  S  to  a 
position  S',  on  the  line  ES' 
drawn  parallel  to  E'S  the 
actual  direction  of  the  sun 
from  the  earth,  and  at  a  dis- 


LAW  OF  THE  ANGULAR  MOTION  OF  A  PLANET.  $3 

tance  ES'  equal  to  E'S  the  actual  distance  of  the  sun  from  the  earth. 
Thus,  for  every  position  of  the  earth  in  its  orbit,  the  corresponding 
apparent  position  of  the  sun  is  obtained  by  drawing  a  line  parallel  to 
the  radius-vector  of  the  earth,  and  equal  to  it.  It  follows,  therefore, 
that  the  area  SES'  apparently  described  by  the  radius-vector  of 
the  sun  (or  the  line  drawn  from  the  sun  to  the  earth)  in  any  inter- 
val of  time,  is  equal  to  the  area  ESE'  actually  described  by  the 
radius-vector  of  the  earth  in  the  same  time ;  and  consequently  that 
the  arc  SS'  apparently  described  by  the  sun  in  space,  is  equal  to 
the  arc  EE'  actually  described  in  the  same  time  by  the  earth. 
Whence  we  conclude,  that  the  apparent  motion  of  the  sun  in  space, 
and  the  actual  motion  of  the  earth,  are  the  same  in  every  particular. 

198,  It  has  been  discovered  that  the  motion  of  the  moon  in  its 
revolution  around  the  earth,  is  subject  to  the  same  laws  as  the  mo- 
tion of  a  planet  in  its  revolution  around  the  sun.  We  shall  assume 
this  to  be  a  fact,  and  show  that  our  hypothesis  is  verified  by  the 
results  to  which  it  leads. 

1 99,  That  point  of  the  orbit  of  a  planet,  which  is  nearest  to  the 
sun,  is  called  the  Perihelion,  and  that  point  which  is  most  distant 
from  the  sun,  the  Aphelion.      The  corresponding  points  of  the 
moon's  orbit,  or  of  the  sun's  apparent  orbit,  are  called,  respective 
ly,  the  Perigee  and  the  Apogee, 

These  points  are  also  called.  Apsides ;  the  former  being  termed 
the  Lower  Apsis,  and  the  latter  the  Higher  Apsis.  The  line  join- 
ing them  is  denominated  the  Line  of  Apsides. 

The  orbits  of  the  sun,  moon,  and  planets,  being  regarded  as  el- 
lipses, the  perigee  and  apogee,  or  the  perihelion  and  aphelion,  are 
the  extremities  of  the  major  axis  of  the  orbit, 

200,  The  law  of  the  angular  motion  of  a  planet  about  the  sun 
may  be   deduced   from    Kepler's  Fig.  42. 

first  law.  Let  PpAp"  (Fig,  42) 
represent  the  orbit  of  a  planet,  con- 
sidered as  an  ellipse,  and  p,  p1  two 
positions  of  the  planet  at  two  in- 
stants separated  by  a  short  interval 
of  fime ;  and  let  n  be  the  middle 
point  of  the  arc  pp1.  With  the  ra- 
dius Sn  describe  the  small  circular 
arc  Inl',  and  with  the  radius  Sb 
equal  to  unity  describe  the  arc  ab. 
It  is  plain  that  the  two  positions  p,p' 
may  be  taken  so  near  to  each  other,  that  the  area  Spp'  will  be 
sensibly  equal  to  the  circular  sector  Sll'.  If  we  suppose  this  to 
be  the  case,  as  the  measure  of  the  sector  is  \lriP  x  Sn  =  \ab  X 
Sn2,  (substituting  for  Inl1  its  value  ab  x  Sn,)  we  shall  have 

area  Sppf  =  ±db  x  Sn2. 
When  the  planet  is  at  any  other  part  of  its  orbit,  as  n',  if 


84  MOTIONS  OF  THE  PLANETS  IN  SPACE. 

Sp"p"  be  an  area  described  in  the  same  time  as  before,  we  shal 
have 


area  Sp"p'"  =  ±a!V  x 
But  these  areas  are  equal  according  to  Kepler's  first  law :  hence, 

±ab  x  Sn*  =  \a!V  xJJM2^.  .  .  (36) ; 
and  ab  :  a'b' :  :  Sri* :  Sn2> 

that  is,  the  angular  motion  of  a  planet  about  the  sun  for  a  short 
interval  of  time,  is  inversely  proportional  to  the  square  of  the  ra- 
dius-vector. 

It  results  from  this  that  the  angular  motion  is  greatest  at  the  pe- 
rihelion, and  least  at  the  aphelion,  and  the  same  at  corresponding 
points  on  either  side  of  the  major  axis  :  also,  that  it  decreases  pro- 
gressively from  the  perihelion  to  the  aphelion,  and  increases  pro- 
gressively from  the  aphelion  to  the  perihelion. 

201.  Now  to  compare  the  true  with  the  mean  angular  motion, 
suppose  a  body  to  revolve  in  a  circle  around  the  sun,  with  the 
mean  angular  motion  of  a  planet,  and  to  set  out  at  the  same  instant 
Fig.  43.  with  it  from  the  perihelion.     Let 

PMAM'  (Fig.  43)  represent  the 
elliptic  orbit  of  the  planet,  and 
PBaB  the  circle  described  by  the 
body.  The  position  B  of  this  fic- 
titious body  at  auy  time  will  be  the 
mean  place  of  the  planet  as  seen 
from  the  sun.  The  two  bodies 
will  accomplish  a  semi-revolution 
in  the  same  period  of  time,  and 
therefore  be,  respectively,  at  A  and 
a  at  the  same  instant ;  for  it  is  ob- 
vious that  the  fictitious  body  will  accomplish  a  semi-revolution  in 
half  the  period  of  a  whole  revolution,  and  by  Kepler's  law  of  areas, 
the  planet  will  describe  a  semi-ellipse  in  half  the  time  of  a  revolu- 
tion. At  the  outset,  the  motion  of  the  planet  is  the  most  rapid, 
(200,)  but  it  continually  decreases  until  the  planet  reaches  the 
aphelion,  while  the  motion  of  the  body  remains  constantly  equal 
to  the  mean  motion.  The  planet  will  therefore  take  the  lead,  and 
its  angular  distance  pSE  from  the  body  will  increase  until  its  mo- 
tion becomes  reduced  to  an  equality  with  the  mean  motion,  after 
which  it  will  decrease  until  the  planet  has  reached  the  aphelion  A, 
where  it  will  be  zero.  In  the  motion  from  the  aphelion  to  the  pe- 
rihelion, the  angular  velocity  of  the  planet  will  at  first  be  less  than 
that  of  the  body,  (200,)  but  it  will  continually  increase,  while 
that  of  the  body  will  remain  unaltered  :  thus,  the  body  will  now 
get  in  advance  of  the  planet,  and  their  angular  distance  p'SB'  will 
increase,  as  before,  until  the  motion  of  the  planet  again  attains  to 
an  equality  with  the  mean  motion,  after  which  it  will  decrease,  as 
before,  until  it  again  becomes  zero  at  the  perihelion. 


DEFINITIONS  OF  TERMS.  85 

It  appears,  then,  that  from  the  perihelion  to  the  aphelion  the 
true  place  is  in  advance  of  the  mean  place,  and  that  from  the  aphe- 
lion to  the  perihelion,  on  the  contrary,  the  mean  place  is  in  ad- 
vance of  the  true  place. 

The  angular  distance  of  the  true  place  of  a  planet  from  its  mean 
place,  as  it  would  be  observed  from  the  sun,  is  called  the  Equa- 
tion of  the  Centre.  Thus,  pSE  is  the  equation  of  the  centre  cor- 
responding to  the  particular  position  p  of  the  planet.  It  is  evident, 
from  the  foregoing  remarks,  that  the  equation  of  the  centre  is  zero 
at  the  perihelion  and  aphelion,  and  greatest  at  the  two  points,  as 
M  and  M',  where  the  planet  has  its  mean  motion.  The  greatest 
value  of  the  equation  of  the  centre  is  called  the  Greatest  Equation 
of  the  Centre. 

202.  As  the  laws  of  the  motion  of  the  moon  (198)  and  of  the 
apparent  motion  of  the  sun  (197)  are  the  same  as  those  of  a  planet, 
the  principles  established  in  the  two  preceding  articles  are  as  ap- 
plicable to  these  bodies  in  their  revolution  around  the  earth,  as  to 
a  planet  in  its  revolution  around  the  sun. 

DEFINITIONS  OF  TERMS. 

203.  (1.)  The  Geocentric  Place  of  a  body  is  its  place  as  seen 
from  the  earth. 

(2.)  The  Heliocentric  Place  of  a  body  is  its  place  as  it  would 
be  seen  from  the  sun. 

(3.)  Geocentric  Longitude  and  Latitude  appertain  to  the  geo- 
centric place,  and  Heliocentric  Longitude  and  Latitude  to  the  he- 
liocentric place. 

(4.)  Two  heavenly  bodies  are  said  to  be  in  Conjunction  when 
their  longitudes  are  the  same,  and  to  be  in  Opposition  when  their 
longitudes  differ  by  180°.  When  any  one  heavenly  body  is  in 
conjunction  with  the  sun,  it  is,  for  the  sake  of  brevity,  said  to  be 
in  Conjunction ;  and  when  it  is  in  opposition  to  the  sun,  to  be  in 
Opposition. 

The  planets  Mercury  and  Venus,  allowing  that  their  distances 
from  the  sun  are  each  less  than  the  earth's  distance  (23),  can  never 
be  in  opposition.  But  they  may  be  in  conjunction,  either  by  being 
between  the  sun  and  earth,  or  by  being  on  the  opposite  side  of  the 
sun.  In  the  former  situation  they  are  said  to  be  in  Inferior  Con- 
junction, and  in  the  latter  in  Superior  Conjunction. 

(5.)  A  Synodic  Revolution  of  a  body  is  the  interval  between 
two  consecutive  conjunctions  or  oppositions. 

For  the  planets  Mercury  and  Venus  a  synodic  revolution  is  the 
interval  between  two  consecutive  inferior  or  superior  conjunctions. 

(6.)  The  Periodic  Time  of  a  planet  is  the  period  of  time  in 
which  it  accomplishes  a  revolution  around  the  sun. 

(7.)  The  Nodes  of  a  planet's  orbit,  or  of  the  moon's  orbit,  are 
tne  points  in  which  the  orbit  cuts  the  plane  of  the  ecliptic.  The 


86 


MOTIONS  OF  THE  PLANETS  IN  SPACE. 


node  at  which  the  planet  passes  from  the  south  to  the  north  side 
of  the  ecliptic  is  called  the  Ascending  Node,  and  is  designated  by 
the  character  &.  The  other  is  called  the  Descending  Node,  and 
is  marked  t3- 

(8.)  The  Eccentricity  of  an  elliptic  orbit  is  the  ratio  which  the 
distance  between  the  centre  of  the  orbit  and  either  focus  bears  to 
the  semi-major  axis. 

Fig.  44 


204.  To  illustrate  these  definitions,  let  EE'E"  (Fig.  44)  repre- 
sent the  orbit  of  the  earth ;  C'DC  the  orbit  of  Venus,  or  Mercury, 
which  we  will  suppose,  for  the  sake  of  simplicity,  to  lie  in  the 
plane  of  the  ecliptic  or  of  the  earth's  orbit ;  LNP  a  part  of  the  or- 
bit of  Mars,  or  of  any  other  planet  more  distant  from  the  sun  S 
than  the  earth  is  ;  and  ANB  a  part  of  the  projection  of  this  orbit 
on  the  plane  of  the  ecliptic :  N  or  &  will  represent  the  ascending 
node  of  the  orbit ;  and  the  descending  node  will  be  diametrically 
opposite  to  this  in  the  direction  Sn'.  Also  let  SV  be  the  direction 
of  the  vernal  equinox,  as  seen  from  the  sun,  and  E  V,  E' V  the  par- 
allel directions  of  the  same  point,  as  seen  from  the  earth  in  the  two 
positions  E  and  E' ;  and  P  being  supposed  to  be  one  position  of 
MarS  in  his  orbit,  let  p  be  the  projection  of  that  position  on  the 
plane  of  the  ecliptic.  The  heliocentric  longitude  and  latitude  of 
Mars  in  the  position  P,  are  respectively  VSp  and  PSp  ;  and  if  the 
earth  be  at  E,  his  geocentric  longitude  and  latitude  are  respec- 
tively VEp  and  PEp.  If  we  suppose  that  when  Mars  is  at  P  the 


ELEMENTS  OF  THE  ORBIT  OF  A  PLANET.  87 

earth  is  at  E',  he  will  be  in  conjunction ;  and  if  we  suppose  the 
earth  to  be  at  E'"  he  will  be  in  opposition.  Again,  if  we  suppose 
the  earth  to  be  at  E,  and  Venus  at  C,  she  will  be  in  superior  con- 
junction; but  if  we  suppose  that  Venus  is. at  C'  at  the  time  that 
the  earth  is  at  E,  she  will  be  in  inferior  conjunction.  The  term 
inferior  is  used  here  in  the  sense  of  lower  in  place,  or  nearer  the 
earth ;  and  superior  in  the  sense  of  higher  in  place,  or  farther  frort 
the  earth.  Since  the  earth  and  planets  are  continually  in  motion, 
it  is  manifest  that  the  positions  of  conjunction  and  opposition  wiL 
recur  at  different  parts  of  the  orbit,  and  in  process  of  time  in  every 
variety  of  position.  The  time  employed  by  a  planet  in  passing 
around  from  one  position  of  conjunction,  or  opposition,  to  another, 
called  the  synodic  revolution,  is,  for  the  same  reason,  longer  than 
the  periodic  time,  or  time  of  passing  around  from  one  point  of  the 
orbit  to  the  same  again. 

if 
ELEMENTS  OF  THE  ORBIT  OF  A  PLANET. 

205.  To  have  a  complete  knowledge  of  the  motions  of  the  plan- 
ets, so  as  to  be  able  to  calculate  the  place  of  any  one  of  them  at 
any  assumed  time,  it  is  necessary  to  know  for  each  planet,  in  ad- 
dition to  the  laws  of  its  motion  discovered  by  Kepler,  the  position 
and  dimensions  of  its  orbit,  its  mean  motion,  and  its  place  at  a  spe- 
cified epoch.     These  necessary  particulars  of  information  are  sub- 
divided into  seven  distinct  elements,  called  the  Elements  of  the 
Orbit  of  a  Planet,  which  are  as  follows  : 

(1.)  The  longitude  of  the  ascending  node. 

(2.)  The  inclination  of  the  plane  of  the  orbit  to  the  plane  of  the 
ecliptic,  called  the  inclination  of  the  orbit. 

(3.)  The  mean  distance  of  the  planet  from  the  sun,  or  the  semi- 
major  axis  of  its  orbit. 

(4.)  The  eccentricity  of  the  orbit. 

(5.)  The  heliocentric  longitude  of  the  perihelion. 

(6.)  The  epoch  of  the  planet  being  at  its  perihelion,  or  instead, 
its  mean  longitude  at  a  given  epoch. 

(7.)  The  periodic  time  of  the  planet. 

The  first  two  ascertain  the  position  of  the  plane  of  the  planet's 
orbit ;  the  third  and  fourth,  the  dimensions  of  the  orbit ;  the  fifth, 
the  position  of  the  orbit  in  its  plane ;  the  sixth,  the  place  of  the 
planet  at  a  given  epoch;  and  the  seventh,  its  mean  rate  of  motion. 

206.  The  elements  of  the  earth's  orbit,  or  of  the  sun's  apparent 
orbit,  are  but  five  in  number ;  the  first  two  of  the  above  -mentioned 
elements  being  wanting,  as  the  plane  of  the  orbit  is  coincident  with 
the  plane  of  the  ecliptic. 

207.  The  elements  of  the  moon's  orbit  are  the  same  with  those 
of  a  planet's  orbit,  it  being  understood  that  the  perigee  of  the  moon's 
orbit  answers  to  the  perihelion  of  a  planet's  orbit,  and  that  the  geo- 
centric longitude  of  the  perigee  arid  the  geocentric  longitude  of  the 


88  MOTIONS  OF  THE  PLANETS  IN  SPACE. 

node  of  the  moon's  orbit  answer,  respectively,  to  the  heliocentric 
longitude  of  the  perihelion  and  the  heliocentric  longitude  of  the 
node  of  a  planet's  orbit. 

208.  The  linear  unit  adopted,  in  terms  of  which  the  semi-major 
axes,  eccentricities,  and  radii-vectores  of  the  planetary  orbits,  are 
expressed,  is  the  mean  distance  of  the  sun  from  the  earth,  or  the 
semi-major  axis  of  the  earth's  orbit.     When  thus  expressed,  these 
lines  are  readily  obtained  in  known  measures  whenever  the  mean 
distance  of  the  sun  becomes  known.     The  lines  of  the  moon's 
orbit  are  found  in  terms  of  the  moon's  mean  distance  from  the 
earth,  as  unity. 

METHODS  OF  DETERMINING  THE  ELEMENTS  OF  THE  SUN'S 
APPARENT  ORBIT,  OR  OF  THE  EARTH'S  REAL  ORBIT. 

MEAN    MOTION. 

209.  The  sun's  mean  daily  motion  in  longitude  results  from  the 
length  of  the  mean  tropical  year  obtained  from  observation,  (192.) 

SEMI-MAJOR  AXIS. 

210.  As  we  have  just  stated,  the  semi-major  axis  of  the  sun's 
apparent  orbit  is  the  linear  unit  in  terms  of  which  the  dimensions 
of  the  planetary  orbits  are  expressed.     Its  absolute  length  is  com- 
puted from  the  mean  horizontal  parallax  of  the  sun. 

211.  The  horizontal  parallax  of  a  body  being  given,  to  find  its 
distance  from  the  earth.     We  have  (equation  7,  p.  51) 

n       R 

~  sin  H  ' 

where  H  represents  the  horizontal  parallax  of  the  body,  D  its  dis- 
tance from  the  centre  of  the  earth,  and  R  the  radius  of  the  earth. 
The  parallax  of  all  the  heavenly  bodies,  with  the  exception  of  the 
moon,  is  so  small,  that  it  may,  without  material  error,  be  taken  in 
this  equation  in  place  of  its  sine.  Thus, 


Again,  since  6.2831853  is  the  length  of  the  circumference  of  a 
circle  of  which  the  radius  is  1,  and  1296000  is  the  number  of 
seconds  in  the  circumference,  we  have  6.2831853  :  1  :  :  1296000"  : 
x  =  206264"  .8  =  the  length  of  the  radius  (1)  expressed  in  seconds. 
Hence,  if  the  value  of  H  be  expressed  in  seconds, 
2068 


212.  In  the  determination  of  the  sun's  parallax,  by  the  process 
of  Arts.  114  and  115,  an  error  of  2"  or  3",  equal  to  about  one- 
fourth  of  the  whole  parallax,  may  be  committed,  so  that  the  dis- 
tance of  the  sun,  as  deduced  by  equation  (38)  from  his  parallax 
found  in  that  manner,  may  be  in  error  by  an  amount  equal  to  one- 


ECCENTRICITY  OF  THE  SUN's  APPARENT  ORBIT.  89 

fourth  or  more  of  the  true  distance.  There  is  a  much  more  ac- 
curate method  of  obtaining  the  sun's  parallax,  which  will  be  no- 
ticed hereafter.  It  has  been  found  by  the  method  to  which  we 
allude,  that  the  horizontal  parallax  of  the  sun  at  the  mean  distance 
is  8".58,  which  may  be  relied  upon  as  exact  to  within  a  small 
fraction  of  a  second.  We  have,  then,  for  the  sun's  mean  distance, 
or  the  mean  semi-major  axis  of  his  orbit, 


D  -  R       .'     =  24040.19  R  =  95,102,992  miles  ; 

o  .5o 

taking  for  R  the  mean  radius  of  the  earth  =  3956  miles. 

ECCENTRICITY. 

213.  First  method.  By  the  greatest  and  least  daily  motions 
in  longitude.  —  Wo  have  already  explained  (194)  the  mode  of  de- 
riving from  observation  the  sun's  motion  in  longitude  from  day  to 
day.  Now,  let  v  =  the  greatest  daily  motion  in  longitude  ;  v'  = 
the  least  daily  motion  in  longitude  ;  r  —  the  least  or  perigean  dis 
tance  of  the  sun  ;  and  r'  the  greatest  or  apogean  distance  ;  and  we 
shall  have,  by  the  principle  of  Art.  200, 

r  :  r'  :  :  V  v'  :  ^  v  ; 
whence,       r'  +  r  :  r'  —  r  :  :  ^  v  -f-  ^  v'  :  ^  v  —  */  v', 

r'  +  r  W+VI7"     ,—       /-7 

or,  -  :  r'  —  r  :  :  --  :  v  v  —  v  v'  : 

2  2 

but, 

r'  +  r 

-  =  semi-major  axis  =  1  ;  and  r'  —  r  =  2  (eccentricity)  =  2  e; 

\/  i)  _i_  \/  <t\i        _         _ 
thus,  1  :  2e  :  :  --  •£  -  :  V.v  —  *J  vf, 

v     7>  -  "*/  1)f 

and  e  =  —L  -  =  .  .  .  (39). 

v  v  +  vV 

The  greatest  and  least  daily  motions  are,  respectively,  (at  a 
mean,)  61  '.165  and  57'.  192.     Substituting,  we  have 
e  =  0.016791. 

The  eccentricity  may  also  be  obtained  from  the  greatest  and 
least  apparent  diameters,  by  a  process  similar  to  the  foregoing,  on 
the  principle  that  the  distances  of  the  sun  at  different  times  are  in- 
versely proportional  to  his  corresponding  apparent  diameters,  (195.) 

214.  Second  method.     By  the  greatest  equation  of  the  centre. 

(1.)  To  find  the  greatest  equation  of  the  centre.  —  Let  L  =  the  true  longitude, 
and  M  =  the  mean  longitude,  at  the  time  the  true  and  mean  motions  are  equal 
between  the  perigee  and  apogee,  (201)  ;  L'  =  the  true  longitude  and  M'  =  the  mean 
longitude,  when  the  motions  are  equal  between  the  apogee  and  perigee  ;  and  E  — 
the  greatest  equation  of  the  centre.  Then  (201) 

L  =  M  +  E,  and  L'  =  M'  —  E  ; 
whence,  L'  -  L  =  M'  -  M  -  2E, 


12 


90  MOTIONS  OP  THE  PLANETS  IN  SPACE. 

About  the  time  of  the  greatest  equation  the  sun's  true  motion,  and  consequently 
the  equation  of  the  centre,  continues  very  nearly  the  same  for  two  or  three  days 
we»may  therefore,  with  but  slight  error,  tako  the  noon,  when  the  sun  is  on  either 
side  of  the  line  of  apsides,  that  separates  the  two  days  on  which  the  motions  in 
longitude  are  most  nearly  equal  to  59'  8",  as  the  epoch  of  the  greatest  equation. 

The  longitude  L  or  L'  at  either  epoch  thus  ascertained,  results  from  the  observed 
right  ascension  and  declination.  M' —  M  =  the  mean  motion  in  longitude  in  the 
interval  of  the  epochs,  and  is  found  by  multiplying  the  number  of  mean  solar  days 
and  fractions  of  a  day  comprised  in  the  interval,  by  59'  8".330,  the  mean  daily  mo- 
tion in  longitude. 

For  example :  from  observations  upon  the  sun,  made  by  Dr.  Maskelyne,  in  the 
year  1775,  it  is  ascertained  in  the  manner  just  explained  that  the  sun  was  near  its 
greatest  equation  at  noon,  or  at  Oh.  3m.  35s.  mean  solar  time,  on  the  2d  April,  and 
at  noon  on  the  31st,  or  at  23h.  49m.  35s.  mean  solar  time,  on  the  30th  of  Septem- 
ber. The  observed  longitudes  were,  at  the  first  period  12°  33'  39".06,  and  at  the 
second  188°  5'  44".45.  The  interval  of  time  between  the  two  epochs  is  182d.  — 
ftm. 

Mean  motion  in  182d.  —  14m.       .     .     .     179°  22'  41".56 
Difference  of  two  longitudes     ....     175    32      5  .39 


Difference 2 )     3     50    36  .17 


Greatest  equation  of  centre       ....         1     55    18  .08 

More  accurate  results  are  obtained  by  reducing  observations  made  during  seve- 
ral days  before  and  after  the  epoch  of  the  greatest  equation,  and  taking  the  mean 
of  the  different  values  of  the  greatest  equation  thus  obtained.  According  to  M 
Delambre,  the  greatest  equation  was  in  1775,  1°  55'  31".66. 

(2.)  The  eccentricity  of  an  orbit  may  be  derived  from  the  greatest  equation  of 
the  centre  by  means  of  the  following  formula  : 

_JK        11  K3        587  K5 
"    2  3^  3.5.216 

Tjl 

in  which  K  stands  for  the  expression  (E  being  the  greatest  equation 

57  . 


of  the  centre.)  In  the  case  of  the  sun's  orbit,  K  being  a  small  fraction,  all  its 
powers  beyond  the  first  may  be  omitted.  Thus,  retaining  only  the  first  term  of  the 
series,  and  taking  E  =  1<>  55'  3l".66  the  greatest  equation  in  1775,  we  have 

K  IP  55'  31".66 

•          6  =  2-  =  2X570.2957795  "  'Ol1  '8°3' 

215.  It  appears  from  the  law  of  the  angular  velocity  of  a  re- 
volving body,  investigated  in  Art.  200,  that  the  amount  of  the  pro- 
portional variation  of  this  velocity,  which  obtains  in  the  course  of 
a  revolution,  depends  altogether  upon  the  amount  of  the  propor- 
tional variation  of  distance,  or,  in  other  words,  upon  the  eccentri- 
city of  the  orbit,  (Def.  8,  p.  86.)  It  follows,  therefore,  that  the 
amount  of  the  greatest  deviation  of  the  true  place  from  the  mean 
place,  that  is,  of  the  greatest  equation  of  the  centre,  (201,)  must 
depend  upon  the  value  of  the  eccentricity.  If  the  eccentricity  be 
great,  the  greatest  equation  of  the  centre  will  have  a  large  value  ; 
and  if  the  eccentricity  be  equal  to  zero,  that  is,  if  the  orbit  be  a 
circle,  the  equation  of  the  centre  will  also  be  equal  to  zero,  or  the 
true  and  mean  place  will  continually  coincide. 

If  either  of  the  two  quantities,  the  greatest  equation  and  the 
eccentricity,  be  known,  the  other,  then,  will  become  determinate  : 
and  formulae  have  been  investigated  which  make  known  either  one 


PERIGEE  OP  THE  SUN*S  APPARENT  ORBIT.  91 

when  the  other  is  given.     Equation  41  is  the  formula  for  the  ec- 
centricity. 

216.  From  observations  made  at  distant  periods,  it  is  discovered 
that  the  equation  of  the  centre,  and  consequently  the  eccentricity, 
is  subject  to  a  continual  slow  diminution.     The  amount  of  the 
diminution  of  the  greatest  equation  in  a  century,  called  the  secular 
diminution,  is  17". 2. 

LONGITUDE  AND  EPOCH  OF  THE  PERIGEE,  j 

217.  As  the  sun's  angular  velocity  is  the  greatest  at  tjie  perigee, 
the  longitude  of  the  sun  at  the  time  its  angular  velocity  (is  greatest, 
will  be  the  longitude  of  the  perigee.     The  time  of  Ae  greatest 
angular  velocity  may  easily  be  obtained  within  a  fewihours,  by 
means  of  the  daily  motions  in  longitude,  derived  from  observation. 

218.  The  more  accurate  method  of  determining  the  longitude 
and  epoch  of  the  perigee,  rests  upon  the  principle  that  the  apogee 
and  perigee  are  the  only  two  points  of  the  orbits  whose  longitudes 
differ  by  180°,  in  passing  from  one  to  the  other  of  which  the  sun 
employs  just  half  a  year.     This  principle  may  be  inferred  from 
Kepler's  law  of  areas,  for  it  is  a  well-known  property  of  the  ellipse, 
that  the  major  axis  is  the  only  line  drawn  through  the  focus  that 
divides  the  ellipse  into  equal  parts,  and  by  the  law  in  question 
equal  areas  correspond  to  equal  times. 

219.  By  a  comparison  of  the  results  of  observations  made  atdis 
tant  epochs,  it  is  discovered  that  the  longitude  of  the  perigee  is 
continually  increasing  at  a  mean  rate  of  61  ".5  per  year.    As  the 
equinox  retrogrades  50". 2  in  a  year,  the  perigee  must  then  have  a 
direct  motion  in  space  of  11". 3  per  year. 

It  will  be  seen,  therefore,  that  the  interval  between  the  times  of 
the  sun's  passage  through  the  apogee  and  perigee,  is  not,  strictly 
speaking,  half  a  sidereal  year,  but  exceeds  this  period  by  the  inter 
val  of  time  employed  by  the  sun  in  moving  through  an  arc  of  5". 6, 
the  sidereal  motion  of  the  apogee  and  perigee  in  half  a  year. 

220.  According  to  the  most  exact  determinations,  the  mean  Ion 
gitude  of  the  perigee  of  the  sun's  orbit  at  the  beginning  of  the  yeai 
1800,  was  279°  30'  8".39  :  it  is  now  280^°. 

221.  The  heliocentric  longitude  of  the  perihelion  of  the  earth's 
orbit,  is  equal  to  the  geocentric  longitude  of  the  perigee  of  the  sun's 
apparent  orbit  minus  180°.  For,  let  AEP  (Fig.  41,  p.  82,)  be  the 
earth's  orbit,  and  PV  the  direction  of  the  vernal  equinox.     When 
the  earth  is  in  its  perihelion  P  the  sun  is  in  its  perigee  S,  and  we 
have  the  heliocentric  longitude  of  the  perihelion  VSP  =  VPL  = 
angle  abc  —  1 80°  =  geocentric  longitude  of  the  sun's  perigee  — 
180°.* 

*  It  is  plain  that  the  same  relation  subsists  between  the  heliocentric  longitude 
of  the  earth  and  the  geocentric  longitude  of  the  sun  in  every  other  position  of  the 
earth  in  its  orbit ;  or  that  each  point  of  the  earth's  orbit  is  diametrically  opposite  tc 
the  corresponding  point  of  the  sun's  apparent  orbit. 


92  MOTIONS  OF  THE  PLANETS  IN  SPACE. 

222.  The  epoch  and  mean  longitude  of  the  perigee  of  the  sun's 
orbit  being  once  found,  the  sun's  mean  longitude  at  any  assumed 
epoch  is  easily  obtained  by  means  of  the  mean  motion  in  longitude. 

METHODS  OF  DETERMINING  THE  ELEMENTS  OF  THE  MOON'S 

ORBIT. 

LONGITUDE  OF  THE  NODE. 

223.  In  order  to  obtain  the  longitude  of  the  moon's  ascending 
node,  we  have  only  to  find  the  longitude  of  the  moon  at  the  time 
its  latitude  is  zero  and  the  moon  is  passing  from  the  south  to  the 
north  side  of  the  ecliptic ;  and  this  may  be  deduced  from  the  lon- 
gitudes and  latitudes  of  the  moon,  derived  from  observed  right  as- 
censions and  declinations  (69),  by  methods  precisely  analogous  to 
those  by  which  the  right  ascension  of  the  sun,  at  the  time  its  decli- 
nation is  zero,  and  it  is  passing  from  the  south  to  the  north  of  the 
equator,  or  the  position  of  the  vernal  equinox,  is  ascertained,  (185.) 

INCLINATION  OF  THE  ORBIT. 

224.  Among  the  latitudes  computed  from  the  moon's  observed 
eight  ascensions  and  declinations,  the  greatest  measures  the  incli- 
nation of  the  orbit.    It  is  found  to  be  about  5°  ;  sometimes  a  little 
greater,  and  at  other  times  a  little  less. 

MEAN  MOTION. 

225.  With  the  longitudes  of  the  moon,  found  from  day  to  day, 
it  is  easy  to  obtain  the  interval  from  the  time  at  which  the  moon 
has  any  given  longitude  till  it  returns  to  the  same  longitude  again. 
This  interval  is  called  a  Tropical  Revolution  of  the  moon.     It  is 
found  to  be  subject  to  considerable  periodical  variations,  and  thus 
one  observed  tropical  revolution  may  differ  materially  from  the 
mean  period f    In  order  to  obtain  the  mean  tropical  revolution,  we 
must  compare  two  longitudes  found  at  distant  epochs.     Their  dif- 
ference, augmented  by  the  product  of  360°  by  the  number  of  rev- 
olutions performed  in  the  interval  of  the  epochs,  will  be  the  mean 
motion  in  longitude  in  the  interval,  from  which  the  mean  motion  in 
100  years  or  36525  days,  called  the  Secular  motion,  may  be  ob- 
tained by  simple  proportion.      The  secular  motion  being  once 
known,  it  is  easy  to  deduce  from  it  the  period  in  which  the  motion 
is  360°,  which  is  the  mean  tropical  revolution. 

It  should  be/ observed,  however,  that  to  find  the  precise  mean  secular  motion  in 
longitude,  it  is  necessary  to  compare  the  mean  longitudes  instead  of  the  true 
Now,  the  true  longitude  of  the  moon  at  any  time  having  been  found,  the  mean 
longitude  at  the  same  time  is  derived  from  it  by  correcting  for  the  equation  of  the 
centre  and  certain  other  periodical  inequalities  of  longitude  hereafter  to  be  noticed. 
But  this  cannot  be  done,  even  approximately,  until  the  theory  of  the  moon's  mo- 
tions is  known  with  more  or  less  accuracy. 

226.  The  longitude  of  the  moon,  at  certain  epochs,  maybe  very 
conveniently  deduced  from  observations  upon  lunar  eclipses.    For, 


MOON'S  MEAN  MOTION  IN  LONGITUDE.  93 

the  time  of  the  middle  of  the  eclipse  is  very  near  the  time  of  oppo- 
sition, when  the  longitude  of  the  moon  differs  180°  from  that  of  the 
sun,  and  the  longitude  of  the  sun  results  from  the  known  theory 
of  its  motion.  The  recorded  observations  of  the  ancients  upon  the 
times  of  the  occurrence  of  eclipses,  are  the  only  observations  that 
can  now  be  made  use  of  for  the  direct  determination  of  the  longi- 
tude of  the  moon  at  an  ancient  epoch. 

227.  The  mean  tropical  revolution  of  the  moon  is  found  to  be 

27.321 582d.  or  27d.  7h.  43m.  4.7s.  (5s.  nearly.) 
Hence,   27.321582d.  :  Id. : :  360°  :  13M7639.  =  13°  10'  35".0  = 
moon's  mean  daily  motion  in  longitude. 

228.  Since  the  equinox  has  a  retrograde  motion,  the  sidereal 
revolution  of  the  moon  must  exceed  the  tropical  revolution,  as  the 
sidereal  year  exceeds  the  tropical  year.    The  excess  will  be  equal 
to  the  time  employed  by  the  moon  in  describing  the  arc  of  preces- 
sion answering  to  a  revolution  of  the  moon.     Thus, 

365.25d. :  50".2  : :  27.3d. :  3". 75  =  arc  of  precession, 
and  13°. 17  :  Id. : :  3".75  :  6.8s.  =  excess. 

Wherefore,  the  mean  sidereal  revolution  of  the  moon  is  27d.  7h. 
43m.  12s. 

229.  It  has  been  found,  by  determining  the  moon's  mean  rate  of  motion  for  pe- 
riods of  various  lengths,  that  it  is  subject  to  a  continual  slow  acceleration.  This 
acceleration  will  not,  however,  be  indefinitely  progressive :  Laplace  has  investiga- 
ted its  physical  cause,  and  shown  from  the  principles  of  Physical  Astronomy,  that 
it  is  really  a  periodical  inequality  in  the  moon's  mean  motion,  which  requires  an 
immense  length  of  time  to  go  through  its  different  values. 

The  mean  motion  given  in  Art.  227  answers  to  the  commencement  of  the  pres- 
ent century. 

LONGITUDE  OF  THE  PERIGEE,  ECCENTRICITY,  AND  SEMI-MAJOR  AXIS. 

230.  The  methods  of  determining  these  elements  of  the  moon's 
orbit  are  similar  to  those  by  which  the  Fig.  45. 

corresponding  elements  of  the    sun's 
orbit  are  found. 

It  is  to  be  observed,  however,  that  for  the 
longitudes  of  the  sun,  which  are  laid  off  in  the 
plane  of  the  ecliptic,  in  the  case  of  the  moon  cor- 
responding angles  are  laid  off  in  the  plane  of  its 
orbit.  These  angles  are  reckoried  from  a  line 
drawn  making  an  angle  with  the  line  of  nodes 
equal  to  the  longitude  of  the  ascending  node,  and 
are  called  Orbit  Longitudes.  The  orbit  longi- 
tude is  equal  to  the  moon's  angular  distance 
from  the  ascending  node  plus  the  longitude  of 
the  ascending  node.  "Thus,  let  VNC  (Fig.  45) 
represent  the  plane  of  the  ecliptic,  and  V'NM  a 
portion  of  the  moon's  orbit  ;  N  being  the  as- 
cending node :  also  let  EV  be  the  direction  of 
the  vernal  equinox,  and  let  EV  be  drawn  in  the 
plane  of  the  moon's  orbit,  making  an  angle 
V'EN  with  the  line  of  {he  nodes  equal  to  YEN, 
the  longitude  of  the  ascending  node  N.  The 
orbit  longitudes  lie  in  the  plane  of  the  moon's  orbit,  and  are  estimated  from  this 
line,  while  the  ecliptic  longitudes  lie  in  the  plane  of  the  ecliptic,  and  are  estimated 


94  MOTIONS  OF  THE  PLANETS  IN  SPACE. 

from  the  line  EV.  Thus,  V'EM,  or  its  measure  V'NM,  is  the  orbit  longitude  of 
the  moon  in  the  position  M ;  and  VEm  is  the  ecliptic  longitude,  that  is,  the  longi- 
tude as  it  has  been  hitherto  considered.  V'NM  =  V'N -f  NM  =  VN  +  NM  ;  that 
is,  orbit  long.  =  long,  of  £^  +  5)'s  distance  from  £^. 

The  orbit  longitudes  are  calculated  from  the  ecliptic  longitudes ;  these  being  de- 
rived from  observed  right  ascensions  and  declinations. 

231.  The  ecliptic  longitude  of  the  moon  at  any  time  being  given,  to  find  the 
orbit  longitude. — As  we  may  suppose  the  longitude  of  the  node  to  be  given,  (223) 
the  equation  of  the  preceding  article  will  make  known  the  orbit  longitude  so  soon 
as  MN,  the  moon's  distance  from  the  node,  becomes  known :  now,  by  Napier's  first 
rule,  we  have 

cos  MNm  =  cot  NM  tang  Nm ; 
or,  cot  NM  =  cos  MNw  cot  Nm. 

Nm  =  ecliptic  long.  —  long,  of  node  ;  and  MNm  =  inclination  of  orbit. 

232.  The  horizontal  parallax  of  the  moon,  like  almost  every 
other  element  of  astronomical  science,  is  subject  to  periodical 
changes  of  value.     It  varies  not  only  during  one  revolution,  but 
also  from  one  revolution  to  another.     The  fixed  and  mean  parallax 
about  which  the  true  parallax  may  be  conceived  to  oscillate,  an- 
swers to  the  mean  distance,  that  is,  the  distance  about  which  the 
true  distance  varies  periodically,  and  is  called  the  Constant  of  the 
Parallax.     It  is,  for  the  equatorial  radius  of  the  earth,  57'  0".9  ; 
from  which  we  find  by  equation  (38)  the  mean  distance  of  the 
moon  from  the  earth  to  be  60.3  radii,  or  about  240,000  miles. 

The  first  equation  of  article  211  would  give  a  more  accurate  result. 
The  greatest  and  least  parallaxes  of  the  moon  are  61'  24"  and  53'  48". 

233.  The  eccentricity  of  the  moon's  orbit  is  more  than  three 
times  as  great  as  that  of  the  sun's  orbit.     Its  greatest  equation  ex- 
ceeds 6°  (215). 

MEAN  LONGITUDE  AT  AN  ASSIGNED  EPOCH. 

234.  We  have  already  explained  (225)  the   principle   of  the 
determination  of  the  mean  longitude   of  the  moon  from  an  ob- 
served true  longitude.     Now,  when  the  mean  longitude   at  any 
one  epoch  whatever  becomes  known,  the  mean  longitude  at  any 
assigned  epoch  is  easily  deduced  from  it  by  means  of  the  mean 
motion  in  longitude. 

METHODS   OF  DETERMINING  THE  ELEMENTS  OF  A  PLANET'S 

ORBIT. 

235.  The  methods  of  determining  the  elements  of  the  planetary 
orbits  suppose  the  possibility  of  finding  the  heliocentric  longitude 
and  the  radius-vector  of  the  earth  for  any  given  time.     Now,  the 
elements  of  the  earth's  orbit  having  been  found  by  the  processes 
heretofore  detailed,  the  longitude  may  be  computed  by  means  of 
Kepler's  first  law,  and  the  radius-vector  from  the  polar  equation 
of  the  elliptic  orbit.     (See  Davies'  Analytical  Geometry,  p.  137.) 
The  manner  of  effecting  such  computation  will  be  considered 


LONGITUDE  OP  THE  NODE  OP  A  PLANET*S  ORBIT. 


95 


hereafter ;  at  present  the  possibility  of  effecting  it  will  be  taken  for 
granted. 

HELIOCENTRIC    LONGITUDE  OP  THE  ASCENDING  NODE. 

236.  When  the  planet  is  in  either  of  its  nodes,  its  latitude  is  zero.  It  follows, 
therefore,  that  the  longitude  of  the  planet  at  the  time  its  latitude  is  zero,  is  the 
geocentric  longitude  of  the  node  at  the  time  the  planet  is  passing  through  it.  Now 
if  the  right  ascension  and  declination  of  the  planet  be  observed  from  day  to  day, 
about  the  time  it  is  passing  from  one  side  of  the  ecliptic  to  the  other,  and  convert- 


ed into  longitude  and  latitude,  the  time  at 
which  the  latitude  is  zero,  and  the  longitude 
at  that  time,  may  be  obtained  by  a  proportion. 
When  the  planet  is  again  in  the  same  node, 
the  geocentric  longitude  of  the  node  may 
again  be  found  in  the  same  manner  as  be. 
fore.  On  account  of  the  different  position 
of  the  earth  in  its  orbit,  this  longitude  will 
differ  from  the  former. 

Now,  if  two  geocentric  longitudes  of  the 
same  node  be  found,  its  heliocentric  longitude 
may  be  computed. — Let  S  (Fig.  46)  be  the 
sun,  N  the  node,  and  E  one  of  the  positions 
of  the  earth  for  which  the  geocentric  longi- 
tude of  the  node  (VEN)  is  known.  Denote 
this  angle  by  G,  the  sun's  longitude  VES  by 
S,  and  the  radius-vector  SE  by  r.  Also,  let 
E'  be  the  other  position  of  the  earth,  and 
denote  the  corresponding  quantities  for  this 
position,  VE'N,  VE'S,  and  SE',  respectively, 
by  G',  S',  and  r1.  Let  the  radius-vector  of 
the  planet  when  in  its  node,  or  SN  =  V  ;  and 
the  heliocentric  longitude  of  the  node,  or  VSN  =  X. 


Fig.  46. 


but 

and 

hence, 

or, 

In  like  manner, 

Dividing, 

rsin  (S  — G) 
°r'  r'sinCS'— GO 
whence, 


The  triangle  'SNE  gives 
sin  SNE  :  sin  SEN  :  :  SE  :  SN  ; 
SEN  =  VES  —  VEN  =  S  —  G, 
SNE  =  VAN  — VSN  =*  VEN  —  VSN  =  G  — X ; 

sin  (G  — X):  sin  (S  — G)::r:  V, 
r  sin  (S  — G)  =  V  sin  (G  — X)  .  .  .  (42). 
r'  sin  (S'  —  G')  =  V  sin  (G'  —X.) 
r  sin  (S  —  G)         sin  (G  —  X) 


r'  sin  (S'  —  G')       sin  (G'  —  X)' 

sin  G  cos  X  —  sin  X  cos  G  sin  G —  cos  G  tang  X 

sin  G'  cos  X  —  sin  X  cos  G'  ~~  sin  G'  — cos  G'tangX  ' 


tangX 


r  sin  (S  —  G)  sin  G'  —  r>  sin  (S'  —  G')  sin  G 

r  sin  (S  —  G)  cos  G'—r1  sin  (S'— G')cosG 


(43). 


Equation  (42)  gives 


r  sin  (S  —  G) 
—  X) 


sin 


(44). 


237.  The  longitude  of  the  node  may  also  be  found  approximately 
from  observations  made  upon  the  planet  at  the  time  of  conjunction 
or  opposition.  It  will  happen  in  process  of  time  that  some  of  the 
conjunctions  and  oppositions  will  occur  when  the  planet  is  near 
one  of  its  nodes ;  the  observed  longitude  of  the  sun  at  this  con- 
junction or  opposition,  will  either  be  approximately  the  heliocentric 


96 


MOTIONS  OF  THE  PLANETS  IN  SPACE. 


Fig.  47. 


E' 


longitude  of  the  node  in  question,  or  will  differ  180°  from  it 
This  will  be  s^een  on  inspecting  Fig.  47.     If  at  a  certain  time  the 

earth  should  be  at  E,  crossing  the 
line  of  nodes,  and  the  planet  in 
conjunction,  it  will  be  in  the  node 
N,  and  VES  the  longitude  of  the 
sun  will  be  equal  to  VSN,  the  heli- 
ocentric longitude  of  the  node.  If 
the  earth  should  be  at  E"  and  the 
planet  in  opposition,  the  longitude  of 
the  sun  would  be  VE"S  -  VE"N 
+  180°  =  VSN  +  180°  =hel.  long, 
of  node  +  180°. 

If  the  daily  variations  of  the  lati- 
tude of  the  planet  should  be  ob- 
served about  the  time  of  the  sup- 
posed conjunction  or  opposition 
near  the  node,  the  time  when  the 
latitude  becomes  zero,  or  the  pla- 
net is  in  its  node,  could  approximately  be  calculated  by  simple 
proportion ;  and  then  so  soon  as  the  rate  of  the  angular  motion 
about  the  sun  becomes  known  (241)  the  longitude  of  the  node 
could  be  more  accurately  determined. 

INCLINATION  OF  THE  ORBIT. 

238.  The  longitude  of  the  node  having  been  found  by  the  pre- 
ceding or  some  other  method,  compute  the  day  on  which  the  sun's 
longitude  will  be  the  same  or  nearly  the  same  :  the  earth  will  then 
be  on  the  line  of  the  nodes.     Observe  on  that  day  the  planet's  right 
ascension  and  decimation,  and  deduce  the  geocentric  longitude  and 
latitude.     Let  ENp  (Fig.  47)  be  the  plane  of  the  ecliptic,  V  the 
vernal  equinox,  S  the  sun,  N  the  node,  E  the  earth  on  the  line  of 
nodes,  and  P  the  planet  as  referred  to  the  celestial  sphere,  from 
the  earth.     Let  X  denote  the  geocentric  latitude  Pp ;  E  the  arc 
Np  =  Vp  — VN  =  geo.  long,  of  planet  —  long,  of  node  ;  and  I 
the  inclination  PNp.     The  right-angled  triangle  PNp  gives 

sin  Np  =  tang  Pp  cot  PNp  =  tang  X  cot  I ; 

.  T      sin  E  T      tang  X 

hence,       cot  I  = -,  and  tang  I  =    .  *      .  .  .  (45) : 

tang  X'  sin  E 

or,  tang  inclination  =  — : — 7= —. — '- — 7- — y-x  . . .  (46). 

sin  (long.  — •  long,  of  node) 

239.  It  will  be  understood,  that  to  obtain  an  exact  result,  we  must  compute  tha 
precise  time  of  the  day  at  which  the  longitude  of  the  sun  is  the  same  as  that  of 
the  node,  and  then,  by  means  of  their  observed  daily  variations,  correct  the  longi- 
tude and  latitude  of  the  planet  for  the  variations  in  the  interval  between  the  tinm 
thus  ascertained  and  the  time  of  the  observation  above  mentioned. 


REDUCTION    OF    OBSERVATIONS.  97 

PERIODIC    TIME. 

240.  The  interval  from  the  time  the  planet  is  in  one  of  its  nodes 
till  its  return  to  the  same,  gives  the  periodic  time  or  sidereal  revo- 
lution. 

241.  Another  and  more  accurate  method  is  to  observe  the  length 
of  a  synodic  revolution,  (p.  85,)  and  compute  the  periodic  time 
from  this.     If  we  compare  the  time  of  a  conjunction  which  has 
been  observed  in  modern  times,  with  that  of  a  conjunction  observed 
by  the  earlier  astronomers,  and  divide  the  interval  between  them 
by  the  number  of  synodic  revolutions  contained  in  it,  we  shall 
have  the  mean  synodic  revolution  with  great  exactness,  from  which 
the  mean  periodic  time  may  be  deduced.* 

The  periodic  time  being  known,  the  mean  daily  motion  around 
the  sun  may  be  found  by  dividing  360°  by  the  periodic  time  ex- 
pressed in  days  and  parts  of  a  day. 

TO  FIND  THE  HELIOCENTRIC  LONGITUDE  AND  LATITUDE,  AND  THE 
RADIUS-VECTOR,  FOR  A  GIVEN  TIME. 

242.  The  earth  being  in  constant  motion  in  its  orbit,  and  being 
thus  at  different  times  very  differently  situated  with  regard  to  the 
other  planets,  as  well  in  respect  to  distance  as  direction,  it  is  ne- 
cessary for  the  purpose  of  comparing  the  observations  made  upon 
these  bodies  with  each  other,  to  refer  them  all  to  one  common 
point  of  observation.     As  the  sun  is  the  fixed  centre  about  which 
the  revolutions  of  the  planets  are  performed,  it  is  the  point  best 
suited  to  this  purpose,  and  accordingly  it  is  to  the  sun  that  the 
observations  are  in  reality  referred.   The  reduction  of  observations 
from  the  earth  to  the  sun,  as  it  is  actually  performed,  consists  in 
the  deduction  of  the  heliocentric  longitude  and  latitude  from  the 
geocentric  longitude  and  latitude,  these  being  derived  from  the 
observed  right  ascension  and  declination. 

We  will  now  show  how  to  effect  this  deduction,  supposing  that  the  longitude  of 
the  node  and  the  inclination  of  the  orbit  are  known.  Let  NP  (Fig.  48)  be  part  of 
the  orbit  of  a  planet,  SNC  the  plane  of  the  ecliptic,  N  the  ascending  node,  S  the 
sun,  E  the  earth,  and  P  the  planet ;  also,  let  P*  be  a  perpendicular  let  fall  from  P 
upon  the  plane  of  the  ecliptic,  and  EV,  SV,  the  direction  of  the  vernal  equinox. 
Let  A  =  PEir  the  geocentric  latitude  of  the  planet ;  /  =  PS*  its  heliocentric  lati- 
tude ;  G  =  VETT  its  geocentric  longitude ;  L  =  VS*  its  heliocentric  longitude  ; 
S  =  VES  the  longitude  of  the  sun  ;  N  =  VSN  the  heliocentric  longitude  of  the 
node  ;  I  =  PNC  the  inclination  of  the  orbit ;  r  =  SE  the  radius- vector  of  the 
earth  ;  and  v  =  SP  the  radius-vector  of  the  planet. 

The  point  *  is  called  the  reduced  place  of  the  planet,  and  S*  its  curtate  distance. 
All  the  angles  of  the  triangle  SEir  have  also  received  particular  appellations  :  SirE 
the  angle  subtended  at  the  reduced  place  of  the  planet  by  the  radius  of  the  earth's 
orbit,  is  called  the  Annual  Parallax,  SB*  the  Elongation,  and  ESr  the  Commu- 

*  We  shall,  in  the  sequel,  investigate  the  equation  that  expresses  the  relation  be- 
tween the  synodic  revolution  and  the  periodic  time.  (See  equation  129,  p.  187)  :  ff 
the  synodic  revolution  («)  be  given,  then,  the  sidereal  year  (P)  being  also  known, 
the  value  of  the  sidereal  revolution  of  the  planet  (j»)  can  be  calculated  from  thia 
equation. 

13 


98 


MOTIONS  OF  THE  PLANETS  IN  SPACE, 
Fig.  48. 


/  V 


tation.    Let  A  =  SirE,  E  =  SE»,  and  C  =ESir.      Draw  Sir  parallel  to  Bi- 
tten A  =  *Sir'  =  VSir  — VS*'  =  VSir  —  VEir  =  L—  G  ;   E  =  VETT  —  VES  = 
G  —  S  ;  C  =  VSE  —  VS*  =  180°  +  VSE'  —  VSff  =  180°  +  VES — VSir  =  180° 
-f  S  —  L  =  T  —  L  (putting  T  =  180°+  S). 

(1.)   For  the  latitude. — The  triangles  EP^,  SPa-,  give 

tang  X       Sir 

Eir  tang  X  =  PJT  =  Sir  tang  /  whence  — — .  =  =-  : 

tang  /       E*  ' 
Sir  _  sin  E  f 
i 


but, 
substituting, 


sin  C 


Sir :  E» :  :  sin  E :  sin  C,  or, 

tang  X  _  sin  E 
tang  /  "~  sin  C ' 
whence,  tang  X  sin  C  =  tang  I  sin  E  .  .  (46) ; 

or,  tang  X  sin  (T—  L)  =  tang  /  sin  (G  —  S)  .  .  .  (47). 

Again,  the  triangle  NP/j  gives,  by  Napier's  first  rule, 

sin  Np  =  cot  PNp  tan  Pp,  or,  sin  (L  —  N)  =  cot  I  tan  I  .  .  (48). 
Either  of  the  equations  (47)  and  (48)  will  give  the  value  of  Z,  when  the  longi- 
tude L  is  known. 

(2.)  For  the  longitude. — If  we  substitute  in  equation  (47)  the  value  of  tang  /, 
given  by  equation  (48),  and  replace  (G  —  S)  by  E,  we  have 

tang  X  sin  (T  —  L)  =  sin  (L  —  N)  tang  I  sin  E  ; 

but  T  — L  =  (T  — N)  — (L  — N)  =  D  — (L— N),  (denoting  (T  — N)   by  D) ; 
substituting,  and  designating  L  —  N  by  a:, 

tang  X  sin  (D  —  x)  =  sin  x  tang  I  sin  E  ; 
whence, 

tang  X  sin  D  cos  x  —  tang  X  cos  D  sin  x  =  tang  I  sin  E  sin  x, 
or,  tang  X  sin  D  —  tang  X  cos  D  tang  x  =  tang  I  sin  E  tang  x, 

which  gives 

tang  X  sin  D 

=  tang  X  cos  D  + tang  I  sin  E   ' 

Substituting  tho  values  of  x,  D,  and  E,  we  have,  finally, 

tangX  sin  (T  — N) 


tang(L—  N)  = 


(50). 


tang  X  cos  (T —  N)  +  tang  I  sin  (G  —  S) 
As  N  is  known,  the  value  of  L  will  result  from  this  equation. 

243.  The  co-oi  linates  employed  to  fix  the  position  of  a  planet 
in  the  plane  of  its  orbit,  are  its  orbit  longitude  (230)  and  its  radius- 
vector,  both  of  which  result  from  the  heliocentric  longitude  and 


REDUCTION  OF  OBSERVATIONS. 


99 


•atitude,  the  longitude  of  the  node  and  the  inclination  of  the  orbit 
being  known. 

In  Fig.  48,  V'NP  represents  the  orbit  longitude,  and  SP  (=s  t>)  the  radius-vector 
for  the  position  P.     Now,  the  triangle  PSir  gives 


and  the  triangle  ES*  gives 

sin  A  :  sin  E  : :  SE  :  Sir 

whence,  by  substitution, 

r  sin  E 


SE  sin  E 

sin  A 


r  sin  E 

sin  A  J 


rsin(G  — S) 


(51). 


sin  A  cos  I      sin  (L  —  G)  cos  I 

The  orbit  longitude  L'  =  NP-f-long.  of  node  .  .  .  (52). 
And  to  find  NP,  the  triangle  NPp  gives 

cos  PNp  =  cot  NP  tang  Np,  or  tang  NP  =  tang  Np  .  .  .  (52) ; 
and  Np  =  long,  of  planet  —  long,  of  node  .  .  .   (52). 

244.  The   heliocentric   longitude  Fig.  49. 
may  be  obtained  in  a  very  simple 

manner,  if  the  observations  be  made 
upon  the  planet  at  the  time  of  con-    c 
junction  or  opposition;  for,  it  will 
then  either  be  equal  to  the  geocen- 
tric longitude,  or  differ  180°  from  it. 

When  the  heliocentric  longitude  is 
thus  found,  the  latitude  for  the  same 
time  may  be  obtained  by  solving  the 
triangle  PNp,  (Fig.  49.)  For,  by  Na- 
pier's first  rule, 

sin  Np  —  cot  PNp  tang  Pp, 

or  tang  Pp  =  sin  Np  tang  PNp  ; 
where  Pp  is  the  latitude  sought, 
PNp  the  known  inclination  of  the  orbit,  and  Np  =  VNp  —  VN  = 
long,  of  planet  —  long,  of  node,  both  of  which  may  be  considered 
as  known. 

The  radius-vector  may  be  computed  for  the  same  time  from 
the  triangle  ESP  ;  for  the  side  SE,  the  radius-vector  of  the  earth, 
is  known,  as  well  as  the  angle  SEP  the  geocentric  latitude  of  the 
planet, andthe  angle  ESP  =  180°  —  PSp  =180°  —  heliocentric lat. 

245.  The  radius- vector  of  either  of  the  inferior  planets  at  the 
time  of  maximum  elongation,  or  greatest  angular  distance  from  the 


sun,  may  be  approximately 
deduced  from  the  amount  of 
the  maximum  elongation,  de- 
termined from  observation. 
The  elongation  which  obtains 
at  any  time  may  be  found  by 
ascertainingfrom  instrumental 
observations* the  places  of  the 


Fig.  50. 


100 


MOTIONS  OF  THE  PLANETS  IN  SPACE. 


planet  and  sun  in  the  heavens,  and  connecting  these  by  an  arc  of  a 
great  circle,  and  with  the  pole  by  other  arcs.  In  the  triangle 
PSp  (Fig.  50)  thus  formed  there  will  be  known  the  two  polar  dis- 
tances PS  and  Pp,  which  are  the  complements  of  the  observed  de- 
cimations, and  the  angle  SPp  the  difference  of  their  observed  right 
ascensions,  from  which  the  angular  distance  Sp  between  the  two 


Fig.  51. 


bodies  may  be  calculated.  The 
maximum  elongation  being, 
then,  supposed  to  be  known, 
let  NPP'  (Fig.  51)  represent 
the  orbit  of  an  inferior  planet. 
The  line  EP  drawn  from  the 
earth  to  the  planet  will,  at  the 
time  of  maximum  elongation, 
be  perpendicular  to  SP  the 
radius-vector  of  the  planet; 
and  thus  we  shall  have  in  the 
right-angled  triangle  EPS,  the 
line  ES,  and  the  angle  SEP, 
from  which  the  radius-vector 
SP  may  be  computed. 

As  the  earth  and  planet  are 
in  motion,  the  greatest  elongation  will  occur  at  different  points  of 
the  planet's  orbit,  and  therefore  we  may  find  by  the  foregoing  pro- 
cess different  radii-vectores. 

LONGITUDE   OF   THE    PERIHELION,  ECCENTRICITY,  AND    SEMI-MAJOR 

AXIS. 

246.  The  longitude  of  the  perihelion, 
the  eccentricity,  and  the  semi-major 
axis,  may  be  derived  from  the  helio- 
centric orbit  longitude  (243)  and  the 
radius-vector  found  for  three  different 
times. 

Let  SP,  SP',  SP"  (Fig.  52)  be  the 
three  given  radii-vectores,  V'SP,  V'SP', 
V'SP",  the  three  given  longitudes,  and 
AB  the  line  of  apsides  of  the  planet's 
orbit.  Let  the  angles  PSF,  PSP", 
which  are  known,  be  represented  by 
m,  n,  and  the  angle  BSP,  which  is 
rf  unknown,  by  x\  and  let  the  three  ra- 
dii-vectores SP,  SP',  SP",  be  denoted  by  v,  t>',  v" ;  the  semi-major  axis  AC  by 
a,  and  the  eccentricity  by  e  :  then,  the  three  unknown  quantities  which  are  to  be 
determined,  are  a,  e,  and  the  angle  x,  and  the  general  polar  equation  of  the  ellipse 
furnishes  for  their  determination  the  three  equations 


1  -f-  e  cos  x 


(53), 


\-\-t  cos  (x  -f-  w») 


1  +  e  cos  (x  -f-  n) 


.  .  .  (55). 


SEMI-MAJOR  AXIS  OF  A  PLANERS  ORBIT.  101 

Equating  the  values  of  a  (1  —  e2)  obtained  from  equations  (53)  and  (54),  we  havt 
v-\-ve  cos  x  =  t>'  -|-  v'e  cos  (x  -{-  m), 

or,  e  =  --  ;  --  -  —  -  —  -  .  .  .  (56). 

v  cos  x  —  v  cos  (x  -f-  »») 

In  like  manner  from  (53)  and  (55), 

v"  —  v 

tjcosa;  —  v"  cos  (x  -}-  ri)  ' 

Let  c'  —  v=p,  and  c"  —  o  =  q  ;  then,  by  equating  the  second  members  of 
equations  (56),  (57),  and  transforming,  we  obtain 

p  _  0  COS  X  -  Vf  COS  (X  -f-  Ml) 

q       v  cos  x  —  1>"  cos  (a;  +  n) 

t)  cos  x  —  v'  cos  m  cos  x  +  »'  sin  m  sin  a; 
c  cos  x  —  v"  cos  n  cos  x  -\-  v"  sin  n  sin  a; 
t>  —  »'  cos  m  -f-  1>'  sin  m  tang  x 
~  v  —  v"  cos  n+  15"  sin  n  tang  a;  ' 

whence,          tang  x  =  '  (°~P//  ,C°Sw)  ~*  (v~V'  C°Sm)  .  .  .  (58). 
qv  sin  m  —  pv   sin  n 

The  value  of  a;  being  found  by  this  equation,  and  subtracted  from  the  orbit  lon- 
gitude of  the  planet  in  the  first  position  P,  the  result  will  be  the  orbit  longitude  of 
the  perihelion.  Also,  x  being  known,  e  may  be  computed  from  either  of  the  equa- 
tions (56)  and  (57)  :  and  hence  again,  the  semi-major  axis  from  equation  (53), 
(54),  or  (55). 

247.  The  semi-major  axis  or  mean  distance  from  the  sun,  may 
also  be  had  by  taking  the  mean  of  a  great  number  of  radii-vectores 
found  for  every  variety  of  position  of  the  planet  in  its  orbit,  (244), 
(245). 

248.  Now  that  Kepler's  third  law  has  been  established  by  in- 
vestigations in  Physical  Astronomy,  it  furnishes  the  most  accurate 
method  of  finding  the  mean  distance  of  a  planet  from  the  sun. 
Thus,  let  P  =  the  periodic  time  of  the  planet,  and  a  =  its  mean 
distance  ;  then,  the  length  of  the  sidereal  year  being  365.256374 
days,  (193), 

(365.256374d.)2:P2::l3:o3; 


249.  If  a  great  number  of  radii-vectores  in  a  great  variety  of  po- 
sitions of  the  planet  in  its  orbit  be  found  by  the  method  explained 
in  Art.  244,  the  longitude  of  the  planet  at  the  time  it  has  the  least 
calculated  radius-vector  will  be  approximately  the  longitude  of  the 
perihelion  :  or,  if  it  chances  that  among  the  radii-vectores  deter- 
mined there  are  two  equal  to  each  other,  the  position  of  the  line 
of  apsides  may  be  found  by  bisecting  the  angle  included  between 
these.     The  ratio  of  the  difference  between  the  greatest  and  least 
calculated  radii-vectores  to  the  mean  of  the  whole,  will  be  the  ap- 
proximate value  of  the  eccentricity. 

EPOCH  OF  A  PLANET  BEING  AT  THE  PERIHELION  OF  ITS  ORBIT. 

250.  From  several  observations  upon  the  planet,  about  the  time 


102  MOTIONS  OF  THE  PLANETS  IN  SPACE. 

it  has  the  same  longitude  as  the  perihelion,  the  correct  time  of  its 
being  at  the  perihelion  may  be  easily  determined  by  proportion. 

251.  The  mean  longitude  at  an  assigned  epoch  is  obtained  up- 
on the  same  principles  as  the  mean  longitude  of  the  sun  or  moon, 
(222,  234.) 

REMARKS. 

252.  The  foregoing  methods  of  determining  the  elements  of  a 
planet's  orbit  suppose  observations  to  be  made  at  two  or  more 
successive  returns  of  the  planet  to  its  node  :  but  it  is  not  necessary 
to  wait  for  the  passage  of  a  planet  through  its  node.     Soon  after 
the  planet  Uranus  was  discovered  by  Sir  William  Herschel,  La- 
place contrived  methods  by  which  the  elements  of  its  elliptic  orbit 
were  determined  from  four  observations  within  little  more  than  a 
year  from  its  first  discovery  by  Herschel.*  After  the  discovery  of 
Ceres,  Gauss  invented  another  general  method  of  calculating  the 
orbit  of  a  planet  from  three  observations,  and  applied  it  to  the  de- 
termination of  the  orbit  of  Ceres,  and,  subsequently,  to  the  deter- 
mination of  the  orbits  of  Pallas,  Juno,  and  Vesta.     This  method 
can  be  more  readily  employed  in  practice  than  that  of  Laplace,  or 
than  any  of  the  solutions  which  other  mathematicians  have  given 
of  the  same  problem,  and  is  now  generally  used  by  astronomers. 

MEAN  ELEMENTS  AND  THEIR  VARIATIONS.  •» 

253.  The  elements  of  the  planetary  orbits,  obtained  by  the  foregoing  processes, 
are  the  true  elements  at  the  periods  when  the  observations  are  made.    Upon  deter- 
mining them  at  different  periods,  it  appears  that  they  are  subject  to  minute  varia- 
tions.   A  comparison  of  the  values  found  at  various  distant  epochs  shows  that  they 
are  slowly  changing  from  century  to  century,  and  that  the  changes  experienced 
during  equal  long  periods  of  time  are  very  nearly  the  same.     The  amount  of  the 
variation  of  an  element  in  a  period  of  100  years  is  called  its  Secular   Variation. 
Upon  reducing  the  elements,  found  at  different  times,  to  the  same  epoch,  by  allow- 
ing for  the  proportional  parts  of  the  secular  variations,  the  different  results  for  each 
element  are  found  to  differ  slightly  from  each  other,  which  shows  that  the  elements 
are  also  subject  to  slight  periodical  variations.  These  variations  being  very  minute, 
the  true  elements  can  never  differ  much  from  the  mean,  or  those  from  which  they 
deviate  periodically  and  equally  on  both  sides. 

The  mean  elements  at  an  assigned  epoch  may  be  had  by  finding  the  true  ele- 
ments at  various  times,  and  reducing  them  to  the  given  epoch,  by  making  allow- 
ance for  the  proportional  parts  of  the  secular  variations,  and  then  taking  for  each 
element  the  mean  of  all  the  particular  values  obtained  for  it. 

254.  A  comparison  of  the  mean  values  of  the  same  element,  found  at  distant 
epochs,  makes  known  the  variation  of  its  mean  value  in  the  interval  between 
them,  from  which  the  secular  variation  may  be  deduced  by  simple  proportion. 

255.  The  elements  of  the  moon's  orbit  are  also  subject  to  continual  variations. 
These  are,  for  the  most  part,  periodic,  and  are  far  greater  than  the  variations  of 
the  corresponding  elements  of  a  planet's  orbit.     It  will  be  seen,  then,  that  in  de- 
termining the  mean  elements,   a  much  greater  number  of  observations  will  be 
required  than  in  the  case  of  a  planetary  orbit.     The  mean  node  and  perigee  have 
a  rapid  and  nearly  uniform  progressive  motion.     Theory  shows  that  the  other 
mean  elements,  with  the  exception  of  the  semi-major  axis,  are  subject  to  secular 
variations,  but  their  effect  has  hitherto  been  very  inconsiderable. 

*  History  of  the  Inductive  Sciences,  vol.  ii.  p.  231. 


ELEMENTS  OF  THE  PLANETARY  ORBITS.  103 

256.  The  mean  elements,  which  have  been  derived  as  above  directly  from  ob- 
servation, have  subsequently  been  verified  and  corrected,  by  comparing  the  com- 
puted with  the  observed  places  of  the  planet ;  and  for  this  purpose  many  thousands 
of  observations  have  been  made. 

257.  Tables  II.  and  III.  contain  the  elements  of  the  orbits  of 
the  principal  planets,  and  of  the  moon's  orbit,  together  with  their 
secular  variations,  for  the  beginning  of  the  year  1801  ;  and  also, 
the  elements  of  the  orbits  of  the  four  small  planets,  Vesta,  Juno, 
Ceres,  and  Pallas,  for  1831.     (See  Note  III.) 

If  an  element  be  desired  for  any  time  different  from  the  epoch 
of  the  table,  we  have  only  to  allow  for  the  proportional  part  of  the 
secular  variation,  in  the  interval  between  the  given  time  and  the 
epoch  of  the  table. 

258.  It  will  be  seen,  on  inspecting  Table  II.,  that  the  mean 
distances  of  the  planets  from  the  sun,  or  the  semi-major  axes  of 
their  orbits,  are  the  only  elements  that  are  invariable.     The  rest 
are  subject  to  minute   secular  variations.     The  nodes  have  all 
retrograde  motions.     The  perihelia,  on  the  contrary,  have  direct 
motions,  with  the  single  exception  of  the  perihelion  of  the  orbit 
of  Venus,  which  has  a  retrograde  motion.     The  eccentricities  of 
some  of  the  orbits  are  increasing,  of  others  diminishing.     That 
of  the  earth's  orbit  is  diminishing. 

The  node  of  the  moon's  orbit  has  a  retrograde  motion,  and  the 
perihelion  a  direct  motion.  The  former  accomplishes  a  tropical 
revolution  in  6788.50982  days,  or  about  18  years  214  days;  and 
the  latter  in  3231.4751  days,  or  in  about  8  years  309  days.  The 
mean  motion  of  the  node,  and  the  mean  motion  of  the  perigee,  are 
both  subject  to  a  slow  secular  diminution. 

259.  It  will  be  seen,  also,  that  the  orbits  of  the  planets  are 
ellipses  of  small  eccentricity,  or  which  differ  but  slightly  from 
circles ;  and  that  they  are,  with  the  exception  of  the  orbit  of 
Pallas,  inclined  under  small  angles  to  the  plane  of  the  ecliptic. 
The  eccentricity  is  in  every  instance  so  small,  that  if  a  represen- 
tation of  the  orbit  were  accurately  delineated,  it  would  not  differ 
perceptibly  from  a  circle.    The  most  eccentric  orbits,  among  those 
of  the  seven  principal  planets,  are  those  of  Mercury  and  Mars ; 
and  the  least  eccentric,  those  of  Venus  and  the  earth.    The  eccen- 
tricity of  Mercury's  orbit  is  12  times  that  of  the  earth's,  of  Mars' 
6  times,  of  Venus'  \.     The  eccentricities  of  the  orbits  of  Jupiter, 
Saturn,  and  Uranus,  are  each  about  3  times  greater  than  that  of 
the  earth's  orbit. 

The  orbit  of  Mercury  is  more  inclined  to  the  ecliptic  than  the 
orbit  of  any  other  of  the  seven  principal  planets  ;  and  the  orbit  of 
Uranus  is  less  inclined  than  that  of  any  other  planet.  The  in- 
clination of  the  latter  is  £ °,  of  the  former  7°. 

The  orbits  of  the  four  asteroids  are  more  eccentric,  and  more 
inclined  to  the  plane  of  the  ecliptic,  than  those  of  the  other  planets 
in  general. 


104  MOTIONS  OF  THE  PLANETS  IN  SPACE. 

260.  The  mean  distances  of  the  planets  from  the  sun  are,  in 
round  numbers,  as  follows  :  Mercury  37  millions  of  miles,  Venus 
69  millions  of  miles,  the  earth  95  millions  of  miles,  Mars  145  mil- 
lions of  miles,  Juno  254  millions  of  miles,  Jupiter  495  millions  of 
miles,  Saturn  907  millions  of  miles,  Uranus  1824  millions  of  miles. 
The  range  of  distance  is  from  1  to  77.     The  distance  of  Uranus 
is  about  19  times  the  earth's  distance:  of  Neptune  30  times. 

261.  The  approximate  periods  of  revolution  of  the  planets  are 
as  follows  :  Mercury  3  months,  (|  of  a  year,)  Venus  7|  months, 
(f  of  a  year,)  Mars  1}  years,  Juno  4f  years,  Vesta  £  of  a  year 
shorter,  and  Ceres  and  Pallas  j  of  a  year  longer  than  that  of 
Juno,  Jupiter  12  years,  (11-f  years,)  Saturn  29|  years,  Uranus 
84  years,  Neptune  164f  years. 

262.  A  remarkable  empirical  law,  called  Bode's  Law  of  the 
Distances,  from  its  discoverer,  the  late  Professor  Bode  of  Berlin, 
connects  the  distances  of  the  planets  from  the  sun.     It  is  as  fol- 
lows.    If  we  take  the  following  numbers,  0,  3,  6,   12,  24,  48, 
96,  192,  and  add  the  number  4  to  each  one  of  them,  so  as  to  ob- 
tain 4,  7,  10,  16,  28,  52,  100,  196,  this  series  of  numbers  will 
express  the  order  of  distance  of  the  planets  from  the  sun.     This 
law  embodies  the  following  curious  relation  between  the  distances 
of  the  orbits  from  one  another,  viz.:  setting  out  from  Venus,  the 
distance  between  two  contiguous  orbits  increases  nearly  in  a  dupli- 
cate ratio  as  we  recede  from  the  sun ;  that  is,  the  distance  from 
the  orbit  of  the  earth  to  the  orbit  of  Mars,  is  twice  the  distance 
from  the  orbit  of  Venus  to  the  orbit  of  the  earth,  and  one  half  the 
distance  from  the  orbit  of  Mars  to  the  orbits  of  the  asteroids,  &c. 
Professor  Challis  of  Cambridge,  England,  has  recently  extended 
this  principle  to  the  distances  of  the  satellites ;  so  that  although 
still  somewhat  indefinite,  it  is  unquestionably  part  of  the  arrange- 
ments and  mechanism  of  the  solar  system.* 

Previous  to  the  discovery  of  the  four  asteroids,  to  complete  the 
above  law  a  planet  was  wanting  between  Mars  and  Jupiter.  It 
was  on  this  account  surmised  by  Bode,  that  another  planet  might 
exist  between  these  two.  Instead  of  one  such  planet,  however,  it 
was  subsequently  discovered  that  there  were  in  fact  four,  revolving 
at  pretty  nearly  the  same  distance  from  the  sun,  and  in  conformity 
with  the  curious  law  which  had  been  detected  by  Bode.  (Note  IV.) 

263.  A  better  idea  of  the  dimensions  of  the  solar  system  than 
is  conveyed  by  the  statement  of  distances  above  given,  may  be 
gained  by  reducing  its  scale  sufficiently  to  bring  it  within  the 
scope  of  familiar  distances.     Thus,  if  we  suppose  the  earth  to  be 
represented  by  a  ball  only  1  inch  in  diameter,  the  distance  of  Mer 
cury  from  the  sun  will  be  represented  on  the  same  scale  by  400  feet, 
the  distance  of  Venus  by  700  feet,  that  of  the  earth  by  1000  feet, 
(j  of  a  mile  nearly,)  that  of  Mars  by  1500  feet,  that  of  Juno  by  £  a 

*  Nichol's  Phenomena  of  the  Solar  System,  p. 


PLACE  OF  A  PLANET  IN  ITS  ORBIT.  105 

mile,  that  of  Jupiter  by  1  mile,  that  of  Saturn  by%2  miles,  (If 
miles,)  and  that  of  Uranus  by  3£  miles,  (3f  miles.)  On  the  same 
scale,  the  distance  of  the  moon  from  the  earth  would  be  only  2^ 
feet :  that  of  Neptune  5|  miles. 


CHAPTER   VIII. 

OP  THE  DETERMINATION  OF  THE  PLACE  OF  A  PLANET,  OR  OF  THE 
SUN,  OR  MOON,  FOR  A  GIVEN  TIME,  BY  THE  ELLIPTICAL  THEORY  J 
AND  OF  THE  VERIFICATION  OF  KEPLER?S  LAWS. 

PLACE  OF  A  PLANET,  OR  OF  THE  SUN,  OR  MOON,  IN  ITS  ORBIT. 

264.  THE  angle  contained  between  the  line  of  apsides  of  a 
planet's  orbit  and  the  radius-vector,  as  reckoned  from  the  peri- 
helion towards  the  east,  is  called  the   True  Anomaly.     Thus, 
let  BPAP'  (Fig.  53)  represent  Fig.  53. 

the  orbit,  B  the  perihelion,  and 
P  the  position  of  the  planet; 
then,  BSP  is  its  true  anomaly. 
The  angle  contained^  between 
the  line  of  apsides  and  the  mean 
place  of  the  planet,  also  reck- 
oned from  the  perihelion  to-  A| 
wards  the  east,  is  called  the 
Mean  Anomaly.  Thus,  let  M  be 
the  mean  place  of  a  planet  at 
the  time  P  is  its  true  place,  and 
BSM  will  be  its  mean  anomaly. 
The  difference  between  the  true  anomaly  BSP  and  the  mean 
anomaly  BSM,  is  the  angular  distance  MSP  between  the  true 
and  mean  place  of  the  planet,  or  the  equation  of  the  centre,  (201.) 

Describe  a  circle  BpA  on  the  line  of  apsides  as  a  diameter ; 
through  P  drawpPD  perpendicular  to  the  line  of  apsides,  and  join 
p  and  C  :  the  angle  BCp,  which  the  line  thus  determined  makes 
with  the  line  of  apsides,  is  called  the  Eccentric  Anomaly. 

The  corresponding  angles  appertaining  to  the  sun's  apparent 
orbit,  and  to  me  moon's  orbit,  have  received  the  same  appellations. 

The  interval  between  two  consecutive  returns  of  a  body  to  either 
apsis  of  its  orbit,  is  called  the  Anomalistic  Revolution.  The  ano- 
malistic revolution  of  the  earth,  or  of  the  sun  in  its  apparent  orbit, 
is  termed,  also,  the  Anomalistic  Year. 

265.  The  periodic  time,  or  the  mean  motion  of  a  body,  and  the 
motion  of  the  apsis  of  its  orbit,  being  known,  the  anomalistic  Devo- 
lution may  l^^asily  computed.     Let  m  =  the  sidereal  motion  of 

S*  14 


108      DETERMINATION  OF  THE  PLACE  OF  A  PLANET. 

the  apsis  answering  to  the  periodic  time,  and  M  =  the  mean  daily 
motion  of  the  planet;  then, 

M  :  Id.  :  :  m  :  x  =  diff.  of  anomalistic  rev.  and  periodic  time. 

When  the  epoch  of  any  one  passage  of  a  planet  through  its 
perihelion,  or  of  the  sun  or  moon  through  its  perigee,  has  been 
found,  we  may,  by  means  of  the  anomalistic  revolution,  deduce 
from  it  the  epoch  of  every  other  passage. 

266.  The  length  of  the  anomalistic  year  exceeds  that  of  the 
sidereal  year  by  4m.  44s. 

267.  From  the  anomalistic  revolution,  and  the  epoch  of  the  last 
passage  through  the  perihelion  or  perigee,  (as  the  case  may  be,) 
we  may  derive  the  mean  anomaly  for  any  given  time.     Let  T  = 
the  anomalistic  revolution,  t  =  the  time  that  has  elapsed  since  the 
last  passage  through  the  perihelion  or  perigee,  and  A  =  the  mean 
anomaly  :  then, 

T  :  360°  :  :  t  :  A  =  360°  -^  .  .  .  (60). 

268.  The  place  of  a  body  in  its  elliptical  orbit  is  ascertained 
by  finding  its  true  anomaly.     The  problem  which  has  for  its  ob- 
ject the  determination  of  the  true  anomaly  from  the  mean,  was  first 
resolved  by  Kepler,  and  is  called  Kepler's  Problem.     Another  and 
more  convenient  method  of  obtaining  the  true  anomaly,  is  to  com- 
pute the  equation  of  the  centre  from  the  mean  anomaly,  and  add 
it  to  the  mean  anomaly,  or  subtract  it  from  it,  according  to  the  po- 
sition of  the  body  in  its  orbit,  (201). 

HELIOCENTRIC  PLACE  OF  A  PLANET. 

269.  The  place  of  a  planet  in  the  plane  of  its  orbit  is  designated 
by  its  orbit  longitude  (230)  and  radius-vector.     To  find  the  orbit 
longitude  we  have  the  equation  V'SP  =  V'SB  +  BSP  (see  Fig. 
53,)  or,         long.  =  long,  of  perihelion  -f  true  anomaly. 

The  orbit  longitude  may  also  be  deduced  from  the  mean  longi- 
tude, by  adding  or  subtracting  the  equation  of  the  centre  ;  for, 
V'SP  -  V'SM  +  MSP, 

or,  true  long.  =  mean  long.  +  equa.  of  centre  : 

also,  V'SP'  =  V'SM'  -  M'SP', 
or,  true  long.  =  mean  long.  —  equa.  of  centre. 

The  radius-vector  results  from  the  polar  equation  of  the  ellip- 
tic orbit,  (235,)  viz  : 


-    e  cos  x 

in  which  x  denotes  the  true  anomaly,  e  the  eccentricity,  and  a  the 
semi-major  axis. 

270.  Now  to  find  the  heliocentric  longitude  and  latitude  which 
ascertain  the  position  of  the  planet  with  respect  to  the  ecliptic,  the 
triangle  NPp  (Fig.  48,  p.  98)  gives 

sin  Pp  =  sin  NP  sin 


GEOCENTRIC  *»LACE  OP  A  PLANET.  107 

or,  sin  lat.  =  sin  (orbit  long.  —  long,  of  node)  x  sin  (inclin.)  .  .  (62); 
and 

cos  PNp  =  tang  Np  cot  NP,  or  tang  Np  =  tang  NP  cos  PNp, 
or, 

tang  (long.  —  long,  of  node)  —  tang  (orbit  long.  —  long,  of  node) 
x  cos  (inclination)  .  .  .  (63). 

GEOCENTRIC  PLACE  OF  A  PLANET 

271.  From  the  heliocentric  longitude  and  latitude  and  the  radius-vector  of  a 
planet,  to  find  the  geocentric  longitude  and  latitude.  —  Let  S  (Fig.  48)  be  the  sun, 
E  the  earth,  P  the  planet,  ir  its  reduced  place,  and  V  the  vernal  equinox.  Denote 
the  heliocentric  longitude  VS*  by  L,  the  heliocentric  latitude  PSir  by  /,  and  the 
radius-vector  SP  by  v  ;  and  denote  the  geocentric  longitude  by  G,  and  the  geo- 
centric latitude  by  A.  Also  let  E  =  SB*  the  elongation  ;  C  =  ES?r  the  commu- 
tation ;  A  =  SffE  the  annual  parallax  ;  and  r  =  SE  the  radius-vector  of  the 
earth.  Now, 

VE*  =  SETT  -f  VES, 
or,  G  =  E  -j-  long,  of  sun. 

This  equation  will  make  known  the  geocentric  longitude  when  the  value  of  E 
is  found.  In  the  triangle  PS*1  the  side  SJT  =  SP  cos  PSn-  =  t>  cos  /,  and  is  there- 
fore known,  the  side  ES  is  given  by  the  elliptical  theory,  (269,)  and  the  angle  C 
may  be  desived  from  the  following  equation  :  C  =  VSE  —  VSjr  =  long,  of  earth 
—  long,  of  planet  ;  and  to  find  E  we  have,  by  Trigonometry, 

ES  +  Sff  :  ES  —  ST  :  :  tan  $  (E^S  -f-  SB*)  :  tan  $  (EWS  —  SE*-,) 
or  r  +  v  cos  I  :  r  —  v  cos  I  :  :  tang  £  (  A  -f-  E)  :  tang  £  (  A  —  E)  ; 

whence, 

v  cos  Z 


tang  4  (A-E)  =  -=  tangi(A+E)  tang  1  ,*+  E) 


Let  tang  6  =  —  -  -  -  :  then, 


tang  J  (A  -  E)  =  ta»g  J  (A  +  E,  ; 


or,  tang  J  (A  —  E)  =  tang  (45°  —  0)  tang  £  (A  +  E)  .  .  .  (64). 

But,  A  +  E  =  180°  —  C,  and  E  =  £  (A  -f  E)  —  £  (A  —  E.) 

Next,  to  find  the  geocentric  latitude. 

Sir  tang  /  =  PJT  =  E*  tang  Aj 

STT        tang  A 

whence,  —  -  =  -  —  —  ?  ; 

EJT        tang  I 

«  •        "El 

but.  Sw  :  Eir  :  :  sin  E  :  sin  C,  or  :=-  =  -  —  —  , 

En-      sin  C 

sin  E       tang  A 

and  therefore  —  —  ^  =  -  —  ^-* 

sin  C       tang  I 

sin  E  tang  I  ,__. 


272.  When  a  planet  is  in  conjunction  or  opposition,  the  sines  of  the  angles  of 
elongation  and  commutation  are  each  nothing.  In  these  cases,  then,  the  geo- 
centric latitude  cannot  be  found  by  the  preceding  formula  ;  it  may,  however,  be 
easily  determined  in  a  different  manner.  Suppose  the  planet  to  be  in  conjunction 
at  P,  (Fig.  49,  p.  99  ;)  then, 

PTT  PIT 


108       DETERMINATION  OF  THE  PLACE  OF  A  PLANET. 

But  the  triangle  SPir  gives 

Par  =  v  sin  Z  and  Sir  =  »  cos  l^  and  ES  =  r ; 

hence,  tang  A  =      *  si"  *        .  .  .  (66)  * 

r  +  v  cos  2 

273.  To  find  the  distance  of  the  planet  from  the  earth,  represent  the  distance  by 
D ;  then,  from  the  triangles  PwS  and  EPrr,  (Fig.  48,  p.  98,)  we  have 

PIT  =  EP  sin  PE?r  =  D  sin  X, 
and  PIT  =  SP  sin  PSs-  =  v  sin  I ; 

whence,  D  =  ^-1  !  .  .  (67). 

sin  A 

274.  The  distance  of  a  planet  being  known,  its  horizontal  parallax  may  be  com- 
puted from  the  equation 

sin  H  =  -,?...  (68.)     (Art.  113.) 


PLACES  OF  THE  SUN  AND  MOON. 

275.  The  place  of  the  sun,  as  seen  from  the  earth,  may  be 
easily  deduced  from  the  heliocentric  place  of  the  earth ;  for  the 
longitude  of  the  sun  is  equal  to  the  heliocentric  longitude  of  the 
earth  plus  180°,  (221,)  and  the  radius-vector  of  the  earth's  orbit  is 
the  same  as  the  distance  of  the  sun  from  the  earth.    But  it  is  more 
convenient  to  regard  the  sun  as  describing  an  orbit  around  the 
earth,  and  to  compute  its  true  anomaly,  (268,)  and  thence  tne  lon- 
gitude and  radius-vector  by  the  equation 

long.  =  true  anomaly  +  long,  of  perigee, 
and  the  polar  equation  of  the  orbit. 

276.  The  orbit  longitude  and  the  radius-vector  of  the  moon  are 
found  by  the  same  process  as  the  longitude  and  radius-vector  of 
the  sun.     The  orbit  longitude  being  known,  the  ecliptic  longitude 
and  the  latitude  may  be  determined  by  a  process  precisely  similar 
to  that  by  which  the  heliocentric  longitude  and  latitude  of  a  plane 
are  found,  (270.) 

VERIFICATION  OF  KEPLER'S  LAWS. 

277.  If  Kepler's  first  two  laws  be  true,  then  the  geocentric 
places  of  the  planets,  computed  by  the  process  that  we  have 
described,  (271^)  which  is  founded  upon  them,  ought  to  agree 
with  the  true  geocentric  places  as  obtained  for  the  same  time  by 
direct  observation ;  or,  the  heliocentric  places  computed  from  the 
observed  geocentric  places,  (242,)  ought  to  agree  with  the  same 
as  computed  by  the  elliptic  theory,  (269,  270.)     Now,  a  great 
number  of  comparisons  have  been  made  between  the  observed  and 
computed  places,  and  in  every  instance  a  close  agreement  between 
the  two  has  been  found  to  subsist.     We  infer,  therefore,  that  the 

*  For  inferior  conjunction  the  sign  of  cos  I  must  be  changed,  and  for  opposition 
the  sign  of  r  must  be  changed. 

•»'•• 


VERIFICATION  OP  KEPLER?S  LAWS.  109 

motions  of  the  planets  must  be  very  nearly  in  conformity  with 
these  laws. 

The  truth  of  the  third  law  has  been  established  by  a  direct 
comparison  of  the  mean  distances  of  the  different  planets  with 
their  periodic  times. 

278.  Kepler's  laws  have  been  verified  for  the  sun  and  moon,  in 
a  similar  manner. 

279.  The  relative  distances  of  the  sun,  or  moon,  at  different 
times,  result  for  this  purpose,  from  measurements  of  the  appa- 
rent diameter,  upon  the  principle  that  any  two  distances  are  in- 
versely proportional  to    the   corresponding  apparent  diameters. 
Let  A  =  semi-diameter  corresponding  to  the  mean  distance,  and 
$  =  semi-diameter  corresponding  to  any  distance  D  :  then 

5  :  A  :  :  1  :  D  ;  whence,  D  =-j-  .  .  .  (69)  ; 

an  equation  which,  when  A  has  been  found,  will  make  known  the 
distance  corresponding  to  any  observed  semi-diameter  <J,  in  terms 
of  the  mean  distance  as  a  unit. 

Now,  to  find  A,  denote  the  greatest  and  least  semi-diameters 
respectively  by  5',  6",  and  the  corresponding  distances  by  D'  and 
D",  and  we  have 

D'=A  D"  =  —  • 

6"  D       <$"' 

and  thence, 

,  or,  != 


2  <5'  <$" 
whence,  A  =  —  ^..  .(70.) 

280.  The  distance  of  the  sun  or  moon  in  terms  of  the  mean 
distance  as  a  unit,  may  be  found  in  this  manner  ;  but  it  may  be 
had  more  accurately  by  means  of  a  principle  which  has  been  dis- 
covered from  observation,  namely,  that  the  distance  is  inversely 
proportional  to  the  square  root  of  the  daily  angular  motion. 


CHAPTER  IX. 

OF  THE  INEQUALITIES  OF  THE  MOTIONS  OF  THE  PLANETS  AND  OF 
THE  MOON  J  AND  OF  THE  CONSTRUCTION  OF  TABLES  FOR  FINDING 
THE  PLACES  OF  THESE  BODIES. 

281.  IT  is  a  general  law  of  nature,  discovered  by  Sir  Isaac 
Newton,  that  bodies  tend,  or  gravitate  towards  each  other,  with 
a  force  directly  proportional  to  their  masses,  and  inversely  pro- 
portional to  the  square  of  their  distance.  The  force  which  causes 
one  body  to  gravitate  towards  another,  is  supposed  to  arise  from  a 


110 


INEQUALITIES  OF  THE  PLANETARY  MOTIONS. 


mutual  attraction  existing  between  the  particles  of  the. two  bodies 
and  is  hence  called  the  Attraction  of  Gravitation.  This  force  of 
attraction,  common  to  all  the  bodies  of  the  Solar  System,  is  the 
general  physical  cause  of  their  motions.  The  sun's  attraction  re- 
tains the  planets  in  their  orbits,  and  the  planets  by  their  mutual 
attractions  slightly  alter  each  other's  motions.  The  reasoning  by 
which  Newton's  Theory  of  Universal  Gravitation  is  established, 
appertains  to  Physical  Astronomy,  and  will  be  presented  in  another 
part  of  the  work. 

282.  If  a  planet  were  acted  on  by  no  other  force  than  the 
attraction  of  the  sun,  it  is  proved  that  its  orbit  would  be  accu- 
rately an  ellipse,  and  that  the  areas  described  by  its  radius-vector 
in  equal  times,  would  be  precisely  equal.     But  it  is  in  reality 
attracted  by  the  other  planets,  as  well  as  the  sun,  and  therefore 
its  actual  motions  cannot  be  in  strict  conformity  with  the  laws 
of  Kepler.     In  fact,  if  we  descend  to  great  accuracy,  the  agree- 
ment between  the  observed  and  computed  places  noticed  in  Art. 
277,  is  found  not  to  be  exact.     The  deviations  from  the  elliptic 
motion,  which  are  produced  by  the  attractions  of  the  planets, 
are  called  Perturbations,  or,  in  Plane  Astronomy,  Inequalities. 
Although,  as  we  have  just  seen,  the  fact  of  the  existence  of  ine- 
qualities in  the  motions  of  the  planets  is  discoverable  from  ob- 
servation, their  laws  cannot  be  determined  without  the  aid  of 
theory. 

283.  In  treating  of  the  perturbations   in  the  motions  of  one 
planet,  resulting  from  the  attractions  of  another,  the  attracting 
planet  is  called  the  Disturbing  Body,  and  the  force  which 


ro- 


Fig.  54. 


this  motion.     This,  then,  is  the  disturbing  force. 


duces  the  perturbations  the 
turbing  Force.  To  find  the  dis- 
turbing force,  let  P  (Fig.  54)  be 
the  planet,  S  the  sun,  and  M  the 
disturbing  body;  and  let  PD 
represent  the  attraction  of  M  for 
the  planet.  Decompose  PD  in- 
to two  forces,  PE  and  PF,  one 
of  which,  PE,  is  equal  arid  paral- 
lel to  SG,  the  attraction  of  M  for 
the  sun ;  the  other,  PF,  will  be 
known  in  position  and  intensity. 
The  two  forces,  PE  and  SG, 
being  equal  and  parallel,  they 
cannot  alter  the  relative  motion 
of  the  sun  and  planet,  and  ac- 
cordingly may  be  left  out  of  ac- 
count :  there  remains,  therefore, 
the  component  PF,  which  will 
be  wholly  effectual  in  disturbing 


PROBLEM  OF  THE  THREE  BODIES.  HI 

It  happens  in  the  case  of  each  planet,  that  the  distances  of  some 
•of  the  other  planets  are  so  great,  that  their  disturbing  forces  are 
insensible.  The  attractions  of  these  bodies  for  the  sun  and  planet, 
when  they  are  exterior  to  the  planet,  are  sensibly  equal  and  paral- 
lel. Owing  to  the  great  distance  of  the  planets  from  each  other 
and  the  smallness  of  their  mass  compared  with  that  of  the  sun,  the 
disturbing  force  is  in  every  instance  very  minute  in  comparison 
with  the  sun's  attraction. 

284.  It  is  plain  that  the  disturbing  force  will,  in  general,  be 
obliquely  inclined  to  the  perpendicular  to  the  plane  of  the  orbit, 
PK,  the  tangent  to  the  orbit,  PT,  and  the  radius- vector,  PS  ;  and 
may,  therefore,  be  decomposed  into  forces  acting  along  these  lines. 
The  component  along  the  perpendicular  will  alter  the  latitude,  and 
the  two  others  both  the  longitude  and  radius-vector ;  that  along 
the  tangent  by  changing  the  velocity  of  the  planet ;  and  that  along 
the  radius-vector  by  changing  the  gravity  towards  the  sun.    It  ap- 
pears, therefore,  that  the  disturbing  force  produces  at  the  same 
time  perturbations  or  inequalities  of  longitude,  of  latitude,  and  of 
radius-vector. 

285.  Let  us  now  consider  how  these  inequalities  may  be  deter- 
mined.    In  the  first  place,  the   inequalities   produced  by  each 

.  disturbing  body,  may  be  separately  investigated  upon  mechanical 
principles,  as  if  the  other  bodies  did  not  exist ;  for  the  reason  that 
the  effect  of  each  disturbing  body  is  sensibly  the  same  that  it  would 
be  if  the  other  bodies  did  not  act.  That  this  is  very  nearly,  if  not 
quite  true,  may  be  at  once  inferred  from  the  minuteness  of  the 
whole  disturbance  produced  by  the  joint  action  of  all  the  disturb- 
ing forces  of  the  system.  1  he  problem  which  has  for  its  object 
the  determination  of  the  inequalities  in  the  motions  of  one  body,  in 
its  revolution  around  a  second,  produced  by  the  attraction  of  a 
third,  is  called  the  Problem  of  the  Three  Bodies.  If,  in  the  case 
of  any  one  planet,  this  problem  be  resolved  for  each  of  the  other 
bodies  of  the  system  which  occasion  sensible  perturbations,  all  the 
inequalities  to  which  the  motion  of  the  planet  is  subject  will  be- 
come known. 

286.  The  general  solution  of  the  problem  of  the  three  bodies, 
that  is,  for  any  mass  and  distance  of  the  disturbing  body,  or  any 
intensity  of  the  disturbing  force,  cannot  be  effected  in  the  existing 
state  of  the  mathematical  sciences.     But  the  problem -has  been 
resolved  for  the  case  that  presents  itself  in  nature,  in  which  the 
disturbing  force  is  very  minute  in  comparison  with  the  central 
attraction.     The  results  obtained  by  the  analysis,  are  certain  an- 
alytical expressions  for  the  perturbations  in  longitude,  latitude,  and 
radius  -vector,  involving  variables  and  constants. 

287.  The  general  expression  for  the  whole  perturbation  in  longitude,  due  to  the 
action  of  any  one  disturbing  body,  is 

•  C  sin  (F  —  P)  -f  C'  sin  2  (F—  P)  +  C"  sin  3  (F  —  P)  -f  &c.  .  .    (71), 

in  which  C,  C',  &c.,  are  constants,  P  the  heliocentric  longitude  of  the  body  di» 


'•  * 

112        INEQUALITIES  OF  THE  PLANETARY  MOTIONS. 

turbed,  and  P'  that  of  the  disturbing  body.  The  number  of  terms  is,  strictly  speak- 
ing, indefinite,  but  they  form  a  decreasing  series,  so  that  only  a  small  number  of 
the  first  terms  (which  will  be  different  in  different  cases)  need  to  be  used. 

288.  The  constants  C,  C',  &c.  are  to  be  determined   from  observation ;  they 
may,  however,  be  determined  in  the  case  of  some  of  the  planets  from  theory  alone. 
The  process  of  finding  them  from  observation  is  as  follows  :  Suppose  that  the  earth 
is  the  body  whose  perturbations  are  under  consideration,  and  let  D  denote  the  per- 
turbation in  longitude,  produced  by  the  joint  action  of  all  the  disturbing  forces. 
Then,  supposing,  for  the  sake  of  simplicity,  that  the  expression  for  the  perturbation 
due  to  each  disturbing  body  consists  of  but  two  terms,  we  have 

D=Csin(F—  P)-f-C'sin2  (P  —  P)-f  csin(F'—  P)+c'sin2  (P"  — P)+&c.  (72). 

Find,  by  observation,  the  heliocentric  longitude  of  the  earth,  (189,  221,)  and  take 
the  difference  between  this  and  the  longitude  as  computed  for  the  same  time  by 
the  elliptical  theory;  (269.)  This  difference  will  be  the  value  of  D  at  the  time  of 
the  observation.  P,  P',  P",  &c.  the  heliocentric  longitudes  of  the  earth  and  of  the 
disturbing  bodies,  and  consequently  P'  —  P,  P"  —  P,  &c.,  are  given  by  the  ellipti- 
cal theory.  Thus,  in  the  above  equation  all  will  be  known  but  C,  C',  c,  c',  &c. 
By  repetitions  of  this  process  as  many  equations  may  be  obtained  as  there  are  con- 
stants to  be  determined,  and  from  these  the  values  of  the  constants  may  be  com- 
puted. But  it  is  usual  to  obtain  a  much  greater  number  of  equations  than  there 
are  constants  ;  as,  by  combining  them  according  to  certain  rules,  much  more  ex- 
act values  of  the  constants  may  be  derived. 

289.  In  the  expression 

C  sin  (F  —  P)  -f-  C'  sin  2  (F  —  P)  -f  &c., 

for  the  perturbation  in  longitude,  due  to  the  action  of  a  disturbing  body,  each  term, 
C  sin  (F—  P),C'  sin  2  (F  — P),  &c.,  is  technically  termed  an  Equation,*  and  is 
considered  as  representing  a  specific  inequality.  The  angle  P'  —  P,  or  2  (P'  —  P), 
or  other  multiple  of  P'  —  P,  the  sine  of  which  enters  into  the  equation  of  an  ine- 
quality, is  called  the  Argument  of  the  inequality ;  and  the  constant  is  called  the 
Coefficient  of  the  inequality.  As  the  greatest  value  of  the  sine  of  the  argument  is 
anity,  the  coefficient  is  equal  to  the  greatest  value  of  the  inequality. 

290.  The  coefficient  being  known,  the  value  of  the  inequality  at  any  particular 
time  will  become  known  if  that  of  the  argument  be  found.     Now,  the  argument 
is  the  difference  between  the  longitudes  of  the  disturbing  body  and  disturbed  body, 
or  some  multiple  of  this  difference,  and  may  be  found  by  the  elliptical  theory.     In 
practice,  the  mean  longitudes  may  be  taken,  without  material  error,  in  place  of  the 
true,  and  these  are  easily  deduced  from  the  mean  longitudes  at  a  given  epoch,  by 
means  of  the  mean  motions  in  longitude  of  the  two  bodies.     When  the  values  of 
all  the  inequalities  in  longitude  have  been  separately  determined,  by  taking  their 
algebraic  sum  yve  shall  have  the  correction  to  be  applied  to  the  elliptic  longitude  in 
order  to  find  the  exact  longitude. 

291.  The  general  expression  for  the  total  perturbation  of  radius- vector,  due  to  the 
action  of  one  body,  is 

C  cos  (F—  P)  +  C'  cos  2  (F  —  P)  +  C"  cos  3  (F  —  P)  -f  &c (73). 

As  in  the  expression  for  the  perturbation  of  longitude,  each  term  is  called  an  equa- 
tion, and  represents  a  distinct  inequality,  the  constant  being  the  coefficient,  and 
the  variable  angle,  the  cosine  of  which  enters  into  the  equation,  the  argument  of 
the  inequality.  The  amounts  of  the  different  inequalities,  at  an  assumed  time, 
are  computed  after  the  same  manner  as  those  of  the  inequalities  of  longitude,  and 
being  added  together  with  their  algebraical  signs,  will  give  the  correction  to  be 
applied  to  the  elliptic  radius-vector. 

292.  The  perturbation  in  latitude  is  very  minute.     The  inequalities  of  latitude, 
as  of  longitude  and  radius- vector,  are  represented  by  equations  composed  of  a  con- 
stant coefficient  and  the  sine  or  cosine  of  a  variable  argument,  or  of  the  form  C 
sin  A  or  C  cos  A. 

» The  term  equation  is  applied  in  Astronomy  to  all  quantities  added  to  mean 
tlements  in  order  to  find  the  true. 


PERIODIC  AND  SECULAR  INEQUALITIES.  113 

293.  The  arguments  of  the  inequalities  we  have  been  consider- 
ing, are  angles  depending  upon  the  configurations  of  the  disturbing 
and  disturbed  planets  with  respect  to  each  other  and  the  sun,  and 
also,  in  some  cases,  with  respect  to  the  nodes  and  perihelia  of  their 
orbits.     Whenever  these  configurations  become  the  same,  as  they 
will  periodically,   the  arguments,   and  therefore  the  inequalities 
themselves,  will  have  the  same  value.     It  follows,  therefore,  that 
the  inequalities  in  question  are  periodic. 

The  interval  of  time  in  which  an  inequality  passes  through  all 
its  gradations  of  positive  and  negative  value,  is  called  the  Period 
of  the  inequality.  It  is  manifestly  equal  to  the  interval  of  time  em- 
ployed by  the  argument  in  increasing  from  zero  to  360°  ;  for,  in 
this  interval  sin  A  or  cos  A  takes  all  its  values,  both  positive  and 
negative,  and  at  the  expiration  of  it  recovers  the  same  value  again. 

294.  It  has  been  stated  that  the  elements  of  the  elliptic  orbits  of 
the  planets  are,  for  the  most  part,  subject  to  a  slow  variation  from 
century  to  century.     Investigations  in  Physical  Astronomy  have 
established  that  the  variations  of  the  elements  are  due  to  the  action 
of  the  disturbing  forces  of  the  planets,  and  that  they  are  not  pro- 
gressive, (except  in  the  cases  of  the  longitude  of  the  node  and  the 
longitude  of  the  perihelion,)  but  are  really  periodic  inequalities 
whose  periods  comprise  many  centuries.     From  the  great  lengths 
of  their  periods  these  inequalities  are  termed  Secular  Inequalities, 
in  order  to  distinguish  them  from  the  inequalities  of  the  elliptic  mo- 
tion, denominated  Periodic  Inequalities,  the  periods  of  which  are 
comparatively  short. 

Physical  Astronomy  furnishes  expressions  called  Secular  Equa- 
tions, which  give  the  value  of  an  element  at  any  assumed  time. 

295.  The  inequalities  of  the  moon's  motion  arise  from  the  dis- 
turbing action  of  the  sun.     The  attractions  of  each  of  the  planets 
for  the  moon  and  earth  are  sensibly  equal  and  parallel.    The  lunar 
inequalities  are  investigated  upon  the  same  principle  as  the  plan- 
etary, and  are  represented  by  equations  of  the  same  general  form, 
that  is,  consisting  of  a  constant  coeificient  and  the  sine  or  cosine 
of  a  variable  argument.     They  far  exceed  in  number  and  magni- 
tude those  of  any  single  planet. 

296.  There  are  three  lunar  inequalities  of  longitude  which  are 
prominent  above  the  rest,  and  were  early  discovered  by  observa-, 
tion. 

The  most  considerable  is  called  the  Evection,  and  was  discover- 
ed by  Ptolemy  in  the  first  century  of  the  Christian  era.  It  has  for 
its  argument  double  the  angular  distance  of  the  moon  from  the 
'Sun  minus  the  mean  anomaly  of  the  moon,  and  amounts  when 
greatest  to  1°  20'  30". 

The  second  is  called  the  Variation,  and  was  discovered  in  the 
sixteenth  century  by  Tycho  Brahe.  Its  argument  is  double  the 
angular  distance  of  the  moon  from  the  sun,  and  its  maximum  value 
is  35'  42". 

15 


1 14        INEQUALITIES  OF  THE  PLANETARY  MOTIONS. 

The  third  is  denominated  the  Annual  Equation,  from  the  cir- 
cumstance of  its  period  being  an  anomalistic  year.     Its  argument 
is  the  mean  anomaly  of  the  sun.    When  greatest  it  amounts  to  11 
12". 

297.  The  discovery  of  the  other  lunar  inequalities  (with  the  ex- 
ception of  one  inequality  of  latitude)  is  due  to  Physical  Astronomy. 

The  whole  number  of  lunar  inequalities  of  longitude,  according 
to  Burckhardt,  is  34,  (without  taking  into  account  the  equation  of 
the  centre  and  the  reduction  from  the  orbit  longitude  to  the  eclip- 
tic longitude  :)  and  according  to  Damoiseau,  45. 

298.  To  present  now  at  one  view  the  entire  process  of  finding 
the  exact  heliocentric  place  of  a  planet,  or  the  geocentric  place  of 
the  moon,  at  any  assumed  time. 

(1.)  Seek  the  elements  of  the  elliptic  orbit  from  a  table  of  ele- 
ments, such  as  Table  II  or  III,  allowing  for  the  proportional  part 
of  the  secular  variation,  or  (more  exactly)  obtain  them  from  their 
secular  equations,  (294.) 

(2.)  Compute  the  longitude,  latitude,  and  radius- vector,  by  the 
elliptic  theory,  (269,  270.) 

(3.)  Compute  the  values  of  the  inequalities  in  longitude  and  lati- 
tude, and  of  radius-vector,  by  means  of  their  equations  (290,  291, 
292,)  and  apply  them  individually  with  their  proper  signs,  as  correc- 
tions to  the  elliptic  values  of  the  longitude,  latitude,  and  radius-vector. 

299.  If  we  suppose  the  sun  to  be  in  motion  instead  of  the  earth, 
its  inequalities  will  be  the  same  as  those  to  which  the  motion  of 
the  earth  is  actually  subject. 

300.  When  the  heliocentric  place  of  a  planet  has  been  found,  its 
geocentric  place,  if  required,  may  be  determined  by  the  process 
explained  in  Art.  271. 

CONSTRUCTION  OF  TABLES. 

301.  The  determination  of  the  place  of  the  sun  or  moon,  or  of  a 
planet,  may  be  greatly  facilitated  by  the  use  of  tables.  -  The  prin- 
ciple and  mode  of  construction  of  tables  adapted  to  this  purpose 
are  nearly  the  same  for  each  body. 

We  will  first  explain  the  mode  of  constructing  tables  for  facilitating  the  com- 
putation of  the  sun's  longitude.  We  have  the  equation 

True  long.  =  mean  long.  -\-  equa.  of  centre  -f-  inequalities  -j-  nutation. 

If,  then,  tables  can  be  constructed  which  will  furnish  by  inspection  the  mean 
longitude,  the  equation  of  the  centre,  the  amounts  of  the  various  inequalities  in 
longitude,  and  the  nutation  in  longitude,  at  any  assumed  time,  we  may  easily  find 
the  true  longitude  at  the  same  time. 

302.  (1.)  For  the  mean  longitude, — The  sun's  mean  motion  in  longitude  in  a 
mean  tropical  year,  is  360°.     From  this  we  may  find  by  proportion  the  mean  mo- 
tions in  a  common  year  of  365  days  and  a  bissextile  year  of  366  days. 

With  these  results,  and  the  mean  longitude  for  the  epoch  of  Jan.  1,  1801,  (see 
Table  II,)  we  may  easily  derive  the  mean  longitude  at  the  beginning  of  each  of 
the  years  prior  and  subsequent  to  the  year  1801.  The  second  column  of  TaWe 
XVIII.  contains  the  mean  longitude  of  the  sun  at  the  beginning  of  each  of  the 
years  inserted  in  the  first  column.  The  third  column  of  this  table  contains  the 


115 

mean  longitude  of  the  perigee  at  the  same  epochs :  it  was  constructed  by  means 
of  the  mean  longitude  of  the  perigee  found  for  the  beginning  of  the  year  1800, 
and  its  mean  yearly  motion  in  longitude,  which  is  61".52.* 

Having  the  sun's  mean  daily  motion  in  longitude,  (192,)  we  obtain  by  propor- 
tion the  motion  in  any  proposed  number  of  months,  days,  hours,  minutes,  or  sec- 
onds. Table  XIX.  contains  the  respective  amounts  of  the  sun's  motion  from  the 
commencement  of  the  year  to  the  beginning  of  each  month ;  Table  XX,  the  sun's 
mean  motion  from  the  beginning  of  any  month  to  the  beginning  of  any  day  of  tho 
month,  and  his  motion  for  hours  from  1  to  24  ;  and  Table  XXI,  the  same  for  min- 
utes  and  seconds  from  1  to  60.  With  these  tables  the  sun's  mean  motion  in  lon- 
gitude in  the  interval  between  any  given  time  in  any  year  and  the  beginning  of  the 
year  may  be  had :  and  if  this  be  added  to  the  epoch  for  the  given  year,  taken  out 
from  Table  XVIII,  the  result  will  be  the  mean  longitude  at  the  given  time.  (See 
Problem  IX.) 

303.  Tables  XIX  and  XX  ateo  contain  the  motions  of  the  sun's  perigee,  from 
which  and  the  epoch  given  by  Table  XVIII  results  the  longitude  of  the  perigee 
at  any  proposed  time.     The  longitude  of  the  perigee  is  given  in  the  Solar  Tables 
for  the  purpose  of  making  known  the  mean  anomaly,  the  mean  anomaly  being 
equal  to  the  mean  longitude  minus  the  longitude  of  the  perigee. 

304.  (2.)  For  the  equation  of  the  centre. — To  find  the  equation  of  the  centre  of 
an  orbit  we  have  the  following  equation  : 

Equa.  of  centre  =  A  sin  6  +  B  sin  20  +  C  sin  3$  +  &c. ; 

in  which  A,  B,  C,  &c.,  are  constants  that  rapidly  decrease  in  value,  and  which 
may  be  determined  for  any  particular  orbit,  and  9  the  mean  anomaly.  Now,  by 
giving  to  the  mean  anomaly  0  in  this  equation  a  series  of  values  increasing  by 
small  equal  differences  (of  1°,  for  instance,)  from  zero  to  360°,  and  computing  the 
corresponding  values  of  the  equation  of  the  centre,  then  registering  in  a  column 
the  different  values  assigned  to  0,  and  in  another  column  to  the  right  of  this  the 
computed  values  of  the  equation  of  the  centre,  we  shall  obtain  a  table  which  will 
give  on  inspection  the  equation  of  the  centre  corresponding  to  any  particular  mean: 
anomaly.  In  this  manner  was  constructed  Table  XXV.  In  this  table,  however, 
for  the  sake  of  compactness,  the  values  of  the  equation,  instead  of  being  register- 
ed in  one  column,  are  put  in  as  many  different  columns  as  there  may  be  different 
numbers  of  signs  in  the  value  of  the  mean  anomaly  ;  each  column  answering  to 
the  particular  number  of  signs  placed  at  the  head  of  it. 

If  the  equation  of  the  centre  at  an  assumed  time  be  required,  find  th,e  mean 
anomaly  by  the  tables  (303,)  and  with  the  value  found  for  it  take  out  the  equation 
of  the  centre  from  Table  XXV. 

The  given  quantity  with  which  a  quantity  is  taken  from  a  table,  is  called  the 
Argument  of.  that  quantity.  Accordingly  the  mean  anomaly  is  the  argument  of 
the  equation  of  the  centre  in  Table  XXV. 

305.  (30  For  the  inequalities. — The  equations  of  the  inequalities,  as  we  have 
already  stated,  are  of  the  form  C  sin  A,  the  argument  A  being  the  difference  be- 
tween the  longitude  of  the  disturbing  planet  and  that  of  the  earth,  or  some  mul- 
liple  of  this  difference.     With  the   equations  of  the  inequalities  a  table  of  each 
inequality  may  be  constructed,  upon  the  same  principles  as  Table  XXV.     But,  as 
the  expression  for  the  whole  perturbation  in  longitude,  (287,)  produced  by  any  one 
.planet,  involves  only  two  variables,  the  longitude  of  the  earth  and  the  longitude  of 
the  planet,  it  is  thought  to  be  more  convenient  to  have  a  table  of  double  entry, 
which  will  £ive  the  amount  of  the  perturbation  by  means  of  the  two  variables  as 
arguments.     Such  a  table  may  be  constructed,  by  assigning  to  the  longitude  of  the 
<jarth  and  the  longitude  of  the  disturbing  planet  a  series  of  values  increasing  by  a 
common  difference,  and  computing  with  each  set  ef  the  values  of  these  quanti- 
ties, the  corresponding  amount  of  the  perturbation. 

In  connection  with  the  tables  of  the  perturbations,  we  must  have  tables  that 
make  known  the  values  of  the  arguments  at  any  given  time.  Now,  the  mean  lon- 
gitude of  the  sun  may  be  found  by  the  solar  tables  (302,)  and  thence  the  mean  he- 


*  The  quantities  in  Table  XVIII  are  called  Epochs.     The  Epoch  of  a  quan*' 
is  its  value  at  some  chosen  epoch. 


116        CONSTRUCTION  OF  ASTRONOMICAL  TABLES. 

liocentric  longitude  of  the  earth  by  subtracting  180°  ;  and  the  mean  longitude  of 
the  disturbing  planet  may  be  had  from  similar  tables.  The  columns  of  Table 
XVIII,  marked  I,  II,  III,  IV,  V,  VI,  VII,  contain  the  arguments  of  all  the  pertur- 
bations, for  the  beginning  of  each  of  the  years  registered  in  the  first  column,  ex 
pressed  in  thousandth  parts  of  a  circle.  Tables  XIX  and  XX  contain  the  varia- 
tions of  the  arguments  for  months,  days,  and  hours.  Their  variations  for  minutes 
and  seccnds  are  too  small  to  be  taken  into  account.  With  these  tables,  and  Table 
XVIII,  the  values  of  the  arguments  at  any  given  time  may  be  found,  and  by 
means  of  the  arguments  the  perturbations  may  be  taken  from  Tables  XXVIII, 
XXIX,  XXX,  XXXI,  XXXII,  and  XXXIII. 

306.  (4.)  For  the  nutation. — The  formula  for  the  lunar  nutation  in  longitude,  is 
17".3  sin  N  —  0".2  sin  2  N,  in  which  N  denotes  the  supplement  (to  360°)  of  the 
longitude  of  the  moon's  ascending  node.    With  this  formula  the  second  column  of 
Table  XXVII  was  constructed.     The  value  of  N,  in  thousandth  parts  of  a  circle, 
results  from  Tables  XVIII,  XIX,  and  XX.     The  solar  nutation  is  also  given  by 
Table  XXVII. 

307.  Tables  may  also  be  constructed  that  will  facilitate  the  computation  of  the 
radius-vector.    We  have 

*   True  rad.  vector  =  elliptic  rad.  vector  +  perturbations. 

A  table  of  the  elliptic  radius-vector  may  be  formed  by  means  of  the  polar  equa- 
tion of  the  orbit,  and  tables  of  the  perturbations  from  their  analytical  expressions, 
(291.)  The  tables  of  the  perturbations  will  have  the  same  arguments  as  the  tables 
of  the  perturbations  of  longitude. 

308.  Lunar  and  planetary  tables  are  constructed  upon  the  same  principles  as  the 
solar  tables  we  have  been  describing,  which  serve  to  make  known  the  orbit  longi- 
tude and  radius- vector.     But  other  tables  are  necessary  in  the  case  of  thes^  bodies, 
for  the  computation  of  the  ecliptic  longitude  and  the  latitude. 

309.  The  difference  between  the  orbit  longitude  and  the  ecliptic  longitude,  is 
called  the  Reduction  to  the  ecliptic.    A  formula  for  the  reduction  has  been  investi- 
gated, in  which  the  variable  is  the  difference  between  the  orbit  longitude  and  the 
longitude  of  the  node,  (or,  what  amounts  to  the  same,  the  orbit  longitude  plus  the 
supplement  of  the  longitude  of  the  node  to  360°.)    If  this  formula  be  reduced  to 
a  table,  by  taking  the  reduction  from  the  table  and  adding  it  to  the  orbit  longitude, 
we  shall  have  the  ecliptic  longitude.  Table  LIII  is  a  table  of  reduction  for  the  moon. 

310.  For  the  latitude,  we  have  the  equation 

True  lat.  =  lat.  in  orbit  +  perturbations. 
We  have  already  seen  (270)  that 

sin  (latin  orbit  )=  sin  (orbit  long.  —  long,  of  node)  sin  ^inclina.) 

A  table  constructed  from  this  formula  will  have  for  its  argument  the  orbit  longi- 
tude minus  the  longitude  of  the  node,  which  is  also  the  argument  of  reduction. 
(See  Table  LV.) 

The  tables  of  the  perturbations  in  latitude  are  constructed  upon  the  same  prin- 
ciples as  the  tables  of  the  perturbations  in  longitude  and  radius-vector. 

311.  A  table  exhibiting  the  longitude  and  latitude,  right  ascen- 
sion and  declination,  distance,  parallax,  semi-diameter,  &c.,  of  the 
sun  or  other  body,  at  stated  periods  of  time,  as  at  noon  of  each  day 
throughout  the  year,  is  called  an  Epliemeris  of  the  body.  An  ephe- 
meris  of  the  sun,  of  the  moon,  and  of  each  of  the  planets,  is  pub- 
lished for  each  year  in  advance  in  the  English  Nautical  Almanac, 
and  in  the  Connaissance  des  Terns. 


FORM  OF  THE  COMETARY  ORBITS.  117 

CHAPTER   X. 

OF  THE  MOTIONS  OF  THE  COMETS. 

312.  WHEN  first  seen,  a  comet  is  ordinarily  at  some  distance 
from  the  sun  in  the  heavens,  and  moving  towards  him.     After 
this  it  continues  to  approach  the  sun  for  a  certain  time,  and  then 
recedes  from  him  to  a  greater  or  less  distance,  and  finally  disap- 
pears.    In  many  instances  comets  have  come  so  near  the  sun,  as 
to  be  for  a  time  lost  in  his  beams.     It  has  sometimes  happened 
that  a  comet  has  not  made  its  appearance  in  the  firmament  until 
after  the  time  of  its  nearest  apparent  approach  to  the  sun,  and 
when  it  is  receding  from  him  in  the  heavens.     This  was  the  case 
with  the  great  comet  of  1843.     It  was  first  seen,  in  this  country, 
in  open  day,  on  the  28th  of  February,  in  the  immediate  vicinity 
(within  3°)  of  the  sun ;  and  after  this  moved  away  from  him,  and 
gradually  diminishing  in  brightness,  in  about  a  month  became 
invisible. 

313.  Comets  resemble  the  planets  in  their  changes  of  apparent 
place  among  the  fixed  stars,  but  they  differ  from  them  in  never 
having  been  observed  to  perform  an  entire  circuit  of  the  heavens. 
Their  apparent  motions  are  also  more  irregular  than  those  of  the 
planets,  and  they  are  confined  to  no  particular  region  of  the  hea- 
vens, but  traverse  indifferently  every  part. 

314.  Sir  Isaac  Newton,  from  observations  that  had  been  made 
upon  the  remarkable  comet  of  1 680,  ascertained  that  this  comet 
described  a  parabolic  orbit,  having  the  sun  at  its  focus,  or  an  ellip- 
tic orbit  of  so  great  an  eccentricity  as  to  be  undistinguishable  from 
a  parabola,  and  that  its  radius-vector  described  equal  areas  in  equal 
times.     Since  then,  the  orbits  of  about  180  comets  have  been 
computed,  and  found  to  be,  with  a  few  exceptions,  of  a  parabolic 
form,  or  sensibly  so. 

315.  It  was  demonstrated  by  Newton,  on  the  theory  of  gravi- 
tation, that  a  body  projected  into  space,  may  describe  about  the 
sun  as  a  focus  either  one  of  the  conic  sections,  and  that  the  form 
of  the  orbit  will  depend  upon  the  projectile  velocity  alone.     With 
one  particular  velocity  the  orbit  will  be  a  parabola  ;  with  any  less 
velocity  it  will  be  an  ellipse  or  circle  ;  and  with  any  greater  velo- 
city it  will  be  an  hyperbola.     Now,  as  there  is  but  one  velocity 
from  which  a  parabolic  orbit  will  result,  and  as  any  comet,  whicn 
may  have  originally  moved  in  an  hyperbola,  must  have  passed 
its  perihelion,  and  receded  beyond  the  limits  of  the  solar  system, 
it  may  be  inferred,  with  great  probability,  that  the  orbits  of  the 
comets  whose  observed  courses  are  not  distinguishable  from  para- 
bolic arcs,  are  in  fact  ellipses  of  great  eccentricity.     This  is  the 
theory  of  the  cometary  motions  proposed  by  Newton. 


118  OF  THE  MOTIONS  OF  THE  COMETS. 

The  orbits  of  some  of  the  comets  are  known  from  observation 
to  be  very  eccentric  ellipses. 

316.  The  elements  of  a  comet's  orbit  are  the  longitude  of  the 
ascending  node,  the  inclination  of  the  orbit,  the  longitude  of  the 
perihelion,  the  perihelion  distance,  and  the  epoch  of  the  perihe- 
lion passage.     These  make  known  the  position  and  dimensions  of 
the  orbit  on  the  supposition  that  it  is  a  parabola,  and  thus  apper- 
tain only  to  the  motions  of  the  comet  for  the  period  during  which 
it  is  visible. 

317.  Assuming  that  the  radius-vector  of  a  comet  describes 
areas  proportional  to  the  times,  the  elements  of  its  orbit  may  be 
computed  from  three  observed  geocentric  places.     But  the  prob- 
lem is  one  of  considerable,  difficulty. 

318.  Astronomers  do  not,  in  general,  seek  to  deduce  from  the 
observations  made  during  one  appearance  of  a  comet  its  entire 
elliptic  orbit.     It  is  impossible,  from  such  observations,  to  com- 
pute the  major-axis  of  its  orbit  and  its  period  with  any  accuracy, 
inasmuch  as  in  the  interval  during  which  they  are  made,  the  comet 
describes  but  a  small  portion  of  its  entire  orbit.     As  examples  of 
the  uncertainty  of  such  determinations,  four  periods  have  been 
found  by  Bessel  for  the  comet  of  1807,  of  which  the  least  rs  1483 
years  and  the  greatest  1952  years  ;  and  that  of  the  great  comet 
seen  in  1811  is  said  to  be  either  2301  or  3056  years.     The  un- 
certainty becomes  rmich  less  when  the  period  of  revolution  is 
short. 

The  only  mode  of  obtaining  the  period  of  a  comet's  revolution 
with  certainty,  is  by  directly  comparing  the  times  of  its  perihelion 
passages.  A  comet  cannot  be  recognised  at  a  second  appearance 
by  its  aspect,  for  this  is  liable  to  great  alterations.  But  it  may  be 
identified  by  means  of  the  elements  of  its  orbit,  as  it  is  extremely 
improbable  that  the  elements  of  the  orbits  of  two  different  comets 
will  agree  throughout.  This  method  of  identifying  a  comet  on  a 
second  appearance  may  sometimes  fail  of  application,  inasmuch 
as  the  orbit  of  a  comet  may  experience  great  alterations,  from  the 
attractions  of  the  planets. 

319.  Owing  to  the  great  lengths  of  the  periods  of  most  of  the 
comets,  and  the  comparatively  short  interval  during  which  their 
motions  have  been  carefully  observed,  there  are  but  three  comets, 
the  periods  and  entire  orbits  of  which  have  been  determined. 
These  are  denominated  Encke's  Comet,  Gambarfs  Comet,  (some- 
times called  Bieltfs,)  and  Halley's  Comet.     The  two  former  have 
never  been  seen,  except  in  a  very  few  instances,  without  the  assist- 
ance of  a  telescope,  but  the  latter,  when  near  its  perihelion,  is  dis- 
tinctly visible  to  the  naked  eye. 

320.  Encke's  Comet  is  so  called  from  Professor  Encke,  of  Ber- 
lin, who  first  ascertained  its  periodical  return..    It  accomplishes 
'ts  revolution  in  the  short  period  of  1207  days,  or  about  3^  years, 
and  moves  in  an  orbit  inclined  under  a  small  angle  (13|°)  to  the 


ENCKE  S  COMET. 


119 


plane  of  the  ecliptic,  and  whose  perihelion  is  at  the  distance  of 
the  planet  Mercury,  and  aphelion  nearly  at  the  distance  of  Jupiter. 

» 
Fig.  55. 


(See  Fig.  55.)  This  discovery  was  made  on  the  occasion  of  its 
fourth  recorded  appearance,  in  1819.  Since  then,  it  has  returned 
several  times  to  its  perihelion,  and  in  every  instance  very  nearly 
as  predicted.  Its  last  return  took  place  in  1848  :  its  next  will  be 
in  March,  1852.  This  comet  is  also  called  the  comet  of  short 
period. 

321.  The  motions  of  this  comet  present  the  anomalous  fact,  in  the  solar  system, 
of  a  period  continually  diminishing,  and  an  orbit  slowly  contracting  from  some 
other  cause  than  the  disturbing  actions  of  the  other  bodies  of  the  system.  Professor 
Encke  finds,  that  after  allowance  has  been  made  for  all  the  perturbations  produced 
by  the  planets,  the  actual  time  of  each  perihelion  passage  anticipates  the  time  cal- 
culated from  the  duration  of  the  previous  revolution  about  2§  hours  ;  and  that  the 
comet  now  arrives  at  its  perihelion  several  days  (about  21)  sooner  than  it  would  if 
the  period  had  remained  unaltered  since  the  comet  was  first  seen,  in  1786.  This 
continual  acceleration  of  the  time  of  the  perihelion  passage  cannot  be  attributed 
to  the  disturbing  attraction  of  some  unknown  body,  because  this  attraction  would 
produce  other  effects,  which  have  not  been  noticed.  Encke  conceives  that  it  can 
arise  from  no  other  cause  than  the  action  of  a  resisting  medium,  or  ether,  in 
space.  The  immediate  effect  of  the  resistance  of  such  a  medium  subsisting  in  the 
regions  of  space  traversed  by  the  comet,  would  be  to  diminish  the  velocity  in 
the  orbit,  which  it  would  at  first  seem  should  delay  the  time  of  the  perihelion 
passage ;  but  the  velocity  being  diminished,  the  centrifugal  force  is  weakened, 
and  consequently,  the  comet  is  drawn  nearer  to  the  sun,  and  moves  in  an  orbit 
lying  within  the  orbit  due  to  the  sun's  attraction  alone :  its  mean  distance  is  there- 
fore diminished,  and  its  period  shortened.  We  have  a  similar  phenomenon  to  this 


, 

120  OF  THE  MOTIONS  OF  THE  COMETS. 

in  the  familiar  fact  of  the  shortening  of  the  arc  of  vibration,  and  consequent  in- 
crease of  the  rapidity  of  vibration  of  a  pendulum,  under  the  influence  of  the  resist- 
ance of  the  air. 

322.  Gambart's  Comet  was  first  seen  by  M.  Biela,  at  Joseph- 
stadt  in  Bohemia,  on  the  27th  of  February,  1826,  and  ten  days  af- 
terwards by  M.  Gambart,  at  Marseilles.     The  latter  calculated  its 
parabolic  elements  from  his  own  observations,  and  on  inspecting 
a  general  table  of  comets  discovered  that  the  same  comet  had  pre- 
viously appeared  in  1805  and  1772.    Its  period  is  about  6f  years, 
(2460  days.)     Its  orbit  is  inclined  under  an  angle  of  13°  to  the 
plane  of  the  ecliptic,  and  has  its  perihelion  just  within  the  orbit  of 
the  earth,  and  aphelion -beyond  the  orbit  of  Jupiter,  (see  Fig.  55.) 
By  a  remarkable  coincidence,  the  orbit  of  this  comet  very  nearly 
intersects  the  orbit  of  the  earth ; — so  nearly  that  if  the  two  bodies 
should  ever  chance  to  arrive  at  the  point  of  crossing  at  the  same 
time,  the  earth  would  encounter  a  portion  of  the  filmy  mass  of  the 
comet.     It  appeared,  according  to  the  prediction,  in  1832;  pass- 
ing through  its  perihelion  on  the  27th  of  November.     At  its  next 
and  last  return,  in  1839,  it  was  not  seen,  owing  to  certain  unfavor- 
able circumstances,  (see  Art.  548.)     It  is  announced  that  it  will 
again  return  to  its  perihelion  on  the  llth  of  February,  1846,  and 
under  favorable  circumstances.    (See  Note  V.) 

Gambart's  comet  and  Encke's  comet  both  have  a  direct  motion, 
or  in  the  order  of  the  signs. 

323.  Halley's  Comet  is  so  called  from  Sir  Edmund  Halley,  se- 
cond Astronomer  Royal  of  England,  who  ascertained  its  period, 
and  correctly  predicted  its  return.     From  a  comparison  of  the  ele- 
ments of  the  orbits  described  by  the  comets  of  1531,  1607,  and 
1682,  he  concluded  that  the  same  comet  had  made  its  appearance 
in  these  several  years,  and  predicted  that  it  would  again  return  to 
its  perihelion  towards  the  end  of  1758  or  the  beginning  of  1759. 
Previous  to  its  appearance  Clairaut,  a  distinguished  French  as- 
tronomer, undertook  the  arduous  task  of  calculating  its  perturba- 
tions from  the  disturbing  actions  of  the  planets  during  this  and  the 
preceding  revolution.     He  found  that  from  this  cause  it  would  be 
retarded  about  618  days  ;  100  days  from  the  eifect  of  Saturn,  and 
518  days  from  the  action  of  Jupiter;  and  predicted  that  it  would 
reach  its  perihelion  within  a  month,  one  way  or  the  other,  of  the 
middle  of  April,  1759.  It  actually  passed  its  perihelion  on  the  12th 
of  March,   1759.     Assuming  the  earth's  mean  distance  from  the 
sun  to  be  unity,  the  perihelion  distance  of  this  comet  is  0.6,  and 
aphelion  distance  35.3.    Accordingly  it  approaches  the  sun  to  with- 
in about  one  half  the  distance  of  the  earth,  and  recedes  from  him 
to  nearly  twice  the  distance  of  Uranus.  (See  Fig.  55.)     Its  period 
is  about  76  years,  but  is  liable  to  a  variation  of  a  year  or  more  from 
the  effect  of  the  attractions  of  the  planets.     The  inclination  of  its 
orbit  is  18°,  and  its  motion  is  retrograde.    The  last  perihelion  pas- 
sage took  place  on  the  16th  of  November,  1835,  within  a  few  days 


GREAT  COMET  OP  1843. 


121 


of  the  predicted  time.  The  next  will  occur  about  the  year  1911. 
It  is  to  be  expected  that  the  perturbations  will  now  be  determined 
with  such  increased  accuracy  that  the  error  in  the  prediction  of 
its  next  perihelion  passage  will  be  less  than  one  day. 

324.  Besides  the  three  comets  whose  motions  have  now  been  described,  there  are 
three  others,  the  orbits  and  periods  of  which  are  supposed  to  be  known,  but  which 
have  not  as  yet  returned  to  verify  the  predictions  concerning  them.  These  are 
Olber's  Comet  of  1815,  the  Great  Comet  of  1843,  and  Faye's  Comet  or  the  third- 
comet  of  1843.  The  first  and  last  are  telescopic  comets.  (See  Note  VI.) 

Fig.  56. 


325.  Olber's  Comet  is  believed  to  accomplish  a  revolution  around  the  sun  in  75 
years  ;  and  to  be  destined  to  return  to  its  perihelion  early  in  the  year  1887. 

326.  The  astronomers  of  the  High  School  Observatory  in  Philadelphia,  and  other 
astronomers  in  Europe,  suppose  that  they  have  identified  the  Great  Comet  of  1843 
with  the  comets  of  1668  and  1689,  and  predict  its  return  about  the  beginning  of 
the  year  1865.    The  probable  identity  of  this  comet  with  that  of  the  year  1668, 

16 


122  OF  THE  MOTIONS  OF  THE  COMETS. 

seems  to  be  generally  admitted  by  astronomers ;  but  more  doubt  is  felt  with  re- 
spect to  the  comet  of  1689.  Professor  Peirce,  of  Harvard  University,  contends 
that  the  arguments  which  have  been  offered  in  support  of  the  identity  of  the  com- 
ets of  1843  and  1689  are  insufficient;  and  finds,  after  an  examination  of  the  dif 
ferent  orbits  which  have  been  calculated,  that  the  observations  are,  on  the  whole, 
best  satisfied  by  the  elliptic  orbit  of  the  French  astronomers  Laugier  and  Mauvais, 
which  answers  to  a  revolution  of  175  years. 

Fig.  56  shows  the  parabolic  path  of  this  comet,  together  with  various  correspond, 
ing  positions  of  the  earth  and  comet,  n  is  the  ascending,  and  n'  the  descending 
node :  the  perihelion,  which  is  within  520,000  miles  of  the  sun's  centre,  is  not  far 
from  midway  between  n  and  n1.  The  inclination  of  the  orbit  is  36°.  The  comet 
passed  its  perihelion  on  the  27th  of  February,  at  about  5  P.  M.,  (Philadelphia 
time.)  On  the  28th  it  was  seen  by  day  at  various  parts  of  New  England,  the 
East  and  West  Indies,  and  the  south  of  Europe.  It  was  then  about  3°  distant 
from  the  sun,  and  of  a  dazzling  brightness.  After  this  it  showed  itself  with  great 
distinctness  early  in  the  evening  over  the  western  horizon ;  and  though  growing 
fainter  from  night  to  night,  as  it  receded  from  the  sun,  continued  visible  to  the 
naked  eye  until  about  the  3d  of  April.  It  was  followed  with  the  telescope  at  the 
High  School  Observatory  until  the  10th  of  April. 

327.  Faye's  Comet  has  a  period  of  only  about  7  years.  Its  perihelion  is  about 
60  millions  of  miles  without  the  earth's  orbit,  and  aphelion  somewhat  beyond  the 
orbit  of  Jupiter.  In  respect  to  eccentricity,  its  orbit  holds  nearly  a  middle  place 
between  those  of  the  two  comets  of  shortest  period  and  the  most  eccentric  planet- 
ary  orbits,  (259.)  The  gradation  is  nearly  as  the  fractions  £,  £,  and  f . 

328.  Of  the  180  comets  whose  paths  have  been  traced,  about 
an  equal  number  have  a  direct  and  a  retrograde  motion.     More 
than  two-thirds  have  the  perihelia  of  their  orbits  within  the  orbit 
of  the  earth.     The  aphelia,  except  in  the  few  instances  already 
cited,  are  beyond  the  orbit  of  Uranus.     Some  have  come  into 
close  proximity  to  the  sun.     The  great  comet  of  1680,  according 
to  the  computation  of  Newton,  came  166  times  nearer  the  sun 
than  the  earth  is.     The  no  less  remarkable  comet  of  1843  seems 
to  have  approached  still  nearer  to  him.    When  at  its  perihelion  it 
was  less  than  100,000  miles  from  the  sun's  surface.     Its  velocity 
at  this  time  was  360  miles  per  second,  and  it  accomplished  a  semi- 
revolution  (from  n  to  n'  in  Fig.  56)  in  the  remarkably  short  inter- 
val of  2  hours.    (See  Note  VII.) 

There  is  little  reason  to  doubt  that  many  of  the  comets  recede 
tens  of  thousands  of  millions  of  miles  from  the  sun  before  they 
begin  to  return  to  him  again.  The  periods  of  most  of  them  are 
told  by  centuries,  and  of  very  many  of  them  by  tens  of  centuries. 
The  planes  of  the  orbits  are  inclined  under  every  variety  of  angle 
to  the  plane  of  the  ecliptic. 

329.  The  motions  of  the  comets  are  liable  to  great  derange- 
ments, from  the  attractions  of  the  planets.     As  their  orbits  cross 
the  orbits  of  the  planets,  they  may  come  into  proximity  to  these 
bodies,  and  be  strongly  attracted  by  them.     Halley's  comet  has 
already  (323)  furnished  an  illustration  of  this  general  fact.     The 
comet  of  1770,  commonly  called  Lexell's  comet,  offers  a  still  more 
striking  example  of  the  disturbances  to  which  the  cometary  motions 
are  exposed.     From  observations  made  upon  this  comet  in  the 
year  1770,  Lexell  made  out  that  its  period  was  5£  years:  still, 
though  a  very  bright  comet,  it  has  not  since  been  seen      Burck. 


i     GREAT  NUMBER  OF  COMETS  123 

hardt  undertook  to  investigate  the  cause  of  this  phenomenon,  and 
found  that,  previous  to  the  year  1767,  the  comet  moved  in  an  orbit 
which  answered  to  a  period  of  50  years,  and  never  approached  near 
enough  to  the  earth  and  sun  to  become  visible.  Early  in  the  year 
1767  it  came  so  near  the  planet  Jupiter  that  his  attraction  changed 
its  orbit  to  one  of  5^  years.  It  thus  became  visible  in  1770,  and 
would  have  again  been  seen  on  its  return  to  the  perihelion  in  1776, 
had  it  not  been  so  situated  with  regard  to  the  earth  and  sun  as  to 
be  continually  hid  by  the  sun's  rays.  In  the  year  1779  it  again 
met  with  Jupiter,  and  passed  so  near  him  that  his  attraction  was 
two  hundred  times  greater  than  the  attraction  of  the  sun.  The 
consequence  was  that  its  orbit  was  greatly  enlarged,  and  its  period 
lengthened  to  20  years  ;  so  that  it  no  longer  comes  near  enough  to 
the  earth  to  be  visible. 

330.  The  number  of  recorded  appearances  of  comets  is  about 
500.  But  the  actual  number  of  cometary  bodies  connected  with 
the  solar  system  is  undoubtedly  far  greater  than  this. 

This  list  comprises  for  the  great  number  of  years  which  precede  the  time  of 
the  invention  of  the  telescope,  only  those  comets  which  were  very  conspicuous  to 
the  naked  eye,  giving,  for  example,  only  three  in  the  thirteenth  and  three  in  the 
fourteenth  century ;  and  since  the  heavens  have  begun  to  be  attentively  examined 
with  telescopes,  from  two  to  three  comets,  on  an  average,  have  made  their  appear- 
ance  every  year,  of  which  the  great  majority  are  telescopic.  The  periods  of  these, 
as  well  as  of  the  others,  are,  in  general,  of  such  vast  length  (328)  that  probably 
not  more  than  half  of  the  whole  number  of  comets  have  returned  twice  to  their 
perihelia  during  the  last  two  thousand  years.  From  these  considerations  it  appears 
that  had  the  heavens  been  attentively  surveyed  with  the  telescope  during  the  last 
two  thousand  years,  as  many  as  2500  different  cometary  bodies  would  have  been 
seen.  But  if  we  reflect  that  there  are  various  causes  which  may  tend  to  prevent 
a  comet  from  being  seen  when  present  in  our  firmament ;  as  unfavorable  weather, 
continued  proximity  to  the  sun,  too  gr»at  distance  from  the  sun  and  earth,  (for  all 
distances  seern  equally  probable,  a  priori.)  want  of  intrinsic  lustre,  (for  there  is  every 
gradation  of  lustre  from  the  highest  to  the  lowest,  and  the  fainter  comets  are  the 
most  numerous,)  &c.,  we  shall  see  it  to  be  highly  probable  that  there  are,  in  fact, 
many  thousands  of  these  bodies.  It  is  not  difficult  to  perceive,  as  Arago  has 
shown,  that  the  paucity  of  observed  comets  with  large  perihelion  distances,  though 
apparently,  is  not  in  fact,  opposed  to  the  natural  supposition  that  the  perihelia  are 
distributed  uniformly  throughout  the  region  of  space  which  surrounds  the  sun,  even 
beyond  the  orbit  of  the  most  distant  planet.  Taking  30  as  the  number  of  comets 
that  come  within  the  orbit  of  Mercury,  this  distinguished  philosopher  finds  that 
upon  this  supposition  with  respect  to  the  distribution  of  the  perihelia,  the  number 
of  comets  which  come  within  the  precincts  of  the  solar  system  is  no  less  than 
three  millions  and  a  half. 

If  the  hypothesis  upon  which  this  estimate  is  based  is  anywhere  near  the  truth, 
then  by  far  the  greater  number  of  the  comets  can  never  be  seen  from  the  earth ; 
Cor  uo  comet  has  ever  been  visible  at  the  distance  of  the  orbit  of  Jupiter. 


124  OF   THE   MOTIONS   OF   THE    SATELLITES. 


CHAPTER    XI. 

OF  THE  MOTIONS  OF  THE  SATELLITES. 

331.  As  it  has  already  been  remarked,  the  planets  which  have 
satellites  are  Jupiter,  Saturn,  and  Uranus.     The  number  of  Jupi- 
ter's satellites  is  four ;  of  Saturn's,  eight ;  of  Uranus',  six. 

332.  The  satellites  of  Jupiter  are  perceptible  with  a  telescope 
of  very  moderate  power.     It  is  found,  by  repeated  observations, 
that  they  are  continually  changing  their  positions  with  respect  to 
one  another  and  the  planet,  being  sometimes  all  to  the  right  of  the 
planet,  and  sometimes  all  to  the  left  of  it,  but  more  frequently 
st>me  on  each  side.     They  are  distinguished  from  each  other  by 
the  distance  to  which  they  recede  from  the  planet,  that  which  re- 
cedes to  the  least  distance  being  called  the  First  Satellite,  that 
which  recedes  to  the  next  greater  distance  the  Second,  and  so  on. 

The  satellites  of  Jupiter  were  discovered  by  Galileo,  in  the 
year  1610. 

333.  The  satellites  of  Saturn  and  of  Uranus  cannot  be  seen 
except  through  excellent  telescopes.     They  experience  changes  of 
apparent  position,  similar  to  those  of  Jupiter's  satellites. 

334.  The  apparent  motion  of  Jupiter's  satellites  alternately  from 
one  side  to  the  other  of  the  planet,  leads  to  the  supposition  that 
they  actually  revolve  around  the  planet.     This  inference  is  con- 
firmed by  other  phenomena.     While  a  satellite  is  passing  from  the 
eastern  to  the  western  side  of  the  planet,  a  small  dark  spot  is  fre- 
quently seen  crossing  the  disc  of  the  planet  in  the  same  direction : 
and  again,  while  the  satellite  is  passing  from  the  western  to  the 
eastern  side,  it  often  disappears,  and  after  remaining  for  a  time 
invisible,   reappears   at   another  place.      These  phenomena  are 
easily  explained,  if  we  suppose  that  the  planet  and  its  satellites 
are  opake  bodies  illuminated  by  the  sun,  and  that  the  satellites  re- 
volve around  the  planet  from  west  to  east.     On  this  hypothesis, 
the  dark  spot  seen  traversing  the  disc  of  the  planet,  is  the  shadow 
cast  upon  it  by  the  satellite  on  passing  between  the  planet  and  the 
sun,  and  the  disappearance  of  the  satellite  is  an  eclipse,  occasioned 
by  its  entering  the  shadow  of  the  planet. 

As  the  transit  of  the  shadow  occurs  during  the  passage  of  the 
satellite  from  the  eastern  to  the  western  side  of  the  planet,  and  the 
eclipse  of  the  satellite  during  its  passage  from  the  western  to  the 
eastern  side,  the  direction  of  the  motion  must  be  from  west  to  east 

335  Analogous  conclusions  may  be  drawn  from  similar  phe- 
nomena exhibited  by  the  satellites  of  Saturn.  The  satellites  of 
Uranus  also  revolve  around  their  primary,  but  the  direction  of  their 
motion,  as  referred  to  the  ecliptic,  is  from  east  to  west. 

336.  Let  us  now  examine  into  the  principal  circumstances  of 


ECLIPSES  OF  JUPITER  S  SATELLITES. 


125 


the  eclipses  of  Jupiter's  satellites,  and  of  the  transits  of  their  shad- 
ows across  the  disc  of  the  primary.  Let  EE'E"  (Fig.  57)  repre- 
sent the  orbit  of  the  earth,  PP'P"  the  orbit  of  Jupiter,  and  ss's" 
that  of  one  of  its  satel-  Fig.  57. 

lites.  Suppose  that  E  is 
the.  position  of  the  earth, 
and  P  that  of  the  planet, 
and  conceive  two  lines, 
aaf,  bb',  to  be  drawn  tan- 
gent to  the  sun  and  plan- 
et :  then,  while  the  satel- 
lite is  moving  from  s  to  s' 
it  will  be  eclipsed,  and 
while  it  is  moving  from 
f  to  f  its  shadow  will 
fall  upon  the  planet. — 
Again,  if  Ee,  Ee'  repre- 
sent two  lines  drawn  from 
the  earth  tangent  to  the 
planet  on  either  side,  the 
satellite  will,  while  mov- 
ing from  g  to  g',  traverse 
the  disc  of  the  planet, 
and  while  moving  from  h 
to  h',  be  behind  the  plan- 
et, and  thus  concealed 
from  view.  It  will  be 
seen  on  an  inspection  of 
the  figure,  .that  during 
the  motion  of  the  earth 
from  E"  the  position  of 
opposition,  to  E'  that  of  conjunction,  the  disappearances  or  immer~ 
sions  of  the  satellite  will  take  place  on  the  western  side  of  the 
planet ;  and  that  the  emersions,  if  visible  at  all,  can  be  so  only 
when  the  earth  is  so  far  from  opposition  and  conjunction  that  the 
line  Es',  drawn  from  the  earth  to  the  point  of  emersion,  will  lie  to 
the  west  of  Ee.  It  will  also  be  seen,  that  during  the  passage  of 
the  earth  from  E'  to  E"  the  emersions  will  take  place  on  the  east- 
ern side  of  the  planet,  and  that  the  immersions  cannot  be  visible, 
unless  the  line  Fs,  drawn  from  the  earth  to  the  point  of  immersion, 
passes  to  the  east  of  the  planet.  It  appears  from  observation  that 
the  immersion  and  emersion  are  never  both  visible  at  the  same  pe- 
riod, except  in  the  case  of  the  third  and  fourth  satellites. 

If  the  orbits  of  the  satellites  lay  in  the  plane  of  Jupiter's  orbit 
an  eclipse  of  each  satellite  would  occur  every  revolution,  but,  in 
point  of  fact,  they  are  somewhat  inclined  to  this  plane,  from  which 
cause  the  fourth  satellite  sometimes  escapes  an  eclipse. 

337.  The  periods  and  other  particulars  of  the  motions  of  the 


126  OP   THE    MOTIONS    OF    THE    SATELLITES. 

satellites,  result  from  observations  upon  their  eclipses.  The  mid- 
dle point  of  time  between  the  satellite  entering  and  emerging  from 
the  shadow  of  the  primary,  is  the  time  when  the  satellite  is  in  the 
direction,  or  nearly  so,  of  a  line  joining  the  centres  of  the  sun  and 
primary.  If  the  latter  continued  stationary,  then  the  interval  be- 
tween this  and  the  succeeding  central  eclipse  would  be  the  periodic 
time  of  the  satellite.  But,  the  primary  planet  moving  in  its  orbit, 
the  interval  between  two  successive  eclipses  is  a  synodic  revolu- 
tion. The  synodic  revolution,  however,  being  observed,  and  the 
period  of  the  primary  being  known,  the  periodic  time  of  the  satel- 
lite may  be  computed. 

338.  The  mean  motions  of  the  satellites  differ  but  little  from 
their  true  motions :  and  hence  the  forms  of  their  orbits  must  be 
nearly  circular.     The  orbit,  however,  of  the  third  satellite  of  Ju- 
piter has  a  small  eccentricity ;  that  of  the  fourth  a  larger. 

339.  The  distances  of  the  satellites  from  their  primary  are  de- 
termined from  micrometrical  measurements  of  their  apparent  dis- 
tances at  the  times  of  their  greatest  elongations. 

A  comparison  of  the  mean  distances  of  Jupiter's  satellites  with 
their  periodic  times,  proves  that  Kepler's  third  law  with  respect  to 
the  planets  applies  also  to  these  bodies ;  or,  that  the  squares  of 
their  sidereal  revolutions  are  as  the  cubes  of  their  mean  distances 
from  the  primary. 

The  same  law  also  has  place  with  the  satellites  of  Saturn  and 
Uranus. 

340.  The  computation  of  the  place  of  a  satellite  for  a  given  time, 
is  effected  upon  similar  principles  with  that  of  the  place  of  a  planet. 
The  mutual  attractions  of  Jupiter's  satellites  occasion  sensible  per- 
turbations of  their  motions,  of  which  account  must  be  taken  when 
it  is  desired  to  determine  their  places  with  accuracy. 

341.  Laplace  has  shown  from  the  theory  of  gravitation,  that,  by 
reason  of  the  mutual  attractions  of  the  first  three  of  Jupiter's  satel- 
lites, their  mean  motions  and  mean  longitudes  are  permanently 
connected  by  the  following  remarkable  relations. 

(1 .)  The  mean  motion  of  the  first  satellite  plus  twice  that  of  the 
third  is  equal  to  three  times  that  of  the  second. 

(2.)  The  mean  longitude  of  the  first  satellite  plus  twice  that  of 
the  third  minus  three  times  that  of  the  second  is  equal  to  180°. 

342.  It  follows  from  this  last  relation,  that  the  longitudes  of  the 
three  satellites  can  never  be  the  same  at  the  same  time,  and  conse- 
quently that  they  can  never  be  all  eclipsed  at  once. 


SOLAR   TIME.  127 

CHAPTER  XTI. 

ON    THE    MEASUREMENT    OF    TIME 

\ 

DIFFERENT  KINDS  OF  TIME. 

843.  IN  Astronomy,  as  we  have  already  stated,  three  kinds  of 
time  are  used — Sidereal,  True  or  Apparent  Solar,  and  Mean 
Solar  Time ;  sidereal  time  being  measured  by  the  diurnal  mo- 
tion of  the  vernal  equinox,  true  or  apparent  solar  time  by  that  of 
the  sun,  and  mean  solar  time  by  that  of  an  imaginary  sun  called 
the  Mean  sun,  conceived  to  move  uniformly  in  the  equator  with 
the  real  sun's  mean  motion  in  right  ascension  or  longitude. 

344.  The  sidereal  day  and  the  mean  solar  day  are  each  of  uni- 
form duration,  but  the  length  of  the  true  solar  day  is  variable,  as 
we  will  now  proceed  to  show. 

The  sun's  daily  motion  in  right  ascension,  expressed  in  time,  is 
equal  to  the  excess  of  the  solar  over  the  sidereal  day.  Now  this 
arc,  and  therefore  the  true  solar  day,  varies  from  two  causes,  viz : 

(1.)   The  inequality  of  the  sun's  daily  motion  in  longitude. 

(2.)   The  obliquity  of  the  ecliptic  to  the  equator. 

If  the  ecliptic  were  coincident  with  the  equator,  the  daily  arc  of 
right  ascension  would  be  equal  to  the  daily  arc  of  longitude,  and 
therefore  would  vary  between  the  limits  57'  11"  and  61'  10", 
which  would  answer,  respectively,  to  the  apogee  and  perigee. 
But,  owing  to  the  obliquity  of  the  ecliptic,  the  inclination  of  the 
daily  arc  of  longitude  to  the  equator  is  subject  to  a  variation  ;  and 
this,  it  is  plain,  (see  Fig.  39,)  will  be  attended  with  a  variation  in 
the  daily  arc  of  right  ascension.  The  tendency  of  this  cause  is 
obviously  to  make  the  daily  arc  of  right  ascension  least  at  the 
equinoxes,  where  the  obliquity  of  the  arc  of  longitude  is  greatest, 
and  greatest  at  the  solstices,  where  the  obliquity  is  least. 

345.  As  the  length  of  the  apparent  solar  day  is  variable,  it 
cannot  conveniently  be  employed  for  the  expression  of  intervals 
of  time ;  moreover,  a  clock,  to  keep  apparent  solar  time,  requires 
to  be  frequently  adjusted.     These  inconveniences  attending  the 
use  of  apparent  solar  time,  led  astronomers  to  devise  a  new 
method  of  measuring  time,  to  which  they  gave  the  name  of 
mean  solar  time.     By  conceiving  an  imaginary  sun  to  move  uni- 
formly in  the  equator  with  the  real  sun's  mean  motion,  a  day  was 
obtained  of  which  the  length  is  invariable,  and  equal  to  the  mean 
length  of  all  the  apparent  solar  days  in  a  tropical  year ;  and  by 
supposing  the  fight  ascension  of  this  fictitious  sun  to  be,  at  the 
instant  of  the  sun's  arrival  at  the  perigee  of  his  orbit,  equal  to  the 
sun's  true  longitude,  and  consequently  at  all  times  equal  to  the 
sun's  mean  longitude,  the  time  deduced  from  its  position  with  re- 


128  MEASUREMENT  OF  TIME. 

spect  to  the  meridian,  was  made  to  correspond  very  nearly  witl. 
apparent  solar  time. 

346.  To  find  the  excess  of  the  mean  solar  day  over  the  sidereal 
day,  we  have  the  proportion 

360°  :  24  sid.  hours  :  :  59'  8".33  :  x  =  3m.  56.555s. 
A  mean  solar  day,  comprising  24  mean  solar  hours,  is,  there- 
fore, 24h.  3m.  56.555s.  of  sidereal  time.     Hence,  a  clock  regula- 
ted to  sidereal  time  will  gain  3m.  56.555s.  in  a  mean  solar  day. 

347.  In  order  to  find  the  expression  for  the  sidereal  day  in 
mean  solar  time,  we  must  use  the  proportion 

24h.  3m.  56.555s.  :  24h.  :  :  24h.  :  x  =  23h.  56m.  4.092s. 
The  difference  between  this  and  24  hours  is  3m.  55.908s. ;  and, 
therefore,  a  mean  solar  clock  will  lose  with  respect  to  a  sidereal 
clock,  or  with  respect  to  the  fixed  stars,  3m.  55.908s.  in  a  sidereal 
day,  and  proportionally  in  other  intervals.  This  is  called  the  daily 
acceleration  of  the  fixed  stars. 

348.  To  express  any  given  period  of  sidereal  time  in  mean  solar  time,  we  must 
subtract  for  each  hour  — '——• -    —  =  9.83s.,  and  for  minutes  and  seconds  in  the 
same  proportion.     And,  on  the  other  hand,  to  express  any  given  period  of  mean 
solar  time  in  sidereal  time,  we  must  add  for  each  hour  —  —  =  9.86s.,  and 

for  minutes  and  seconds  in  the  same  proportion. 

349.  It  is  the  practice  of  astronomers  to  adjust  the  sidereal  clock  to  the  motions 
of  the  true  instead  of  the  mean  equinox.     The  inequality  of  the  diurnal  motion  of 
this  point  is  too  small  to  occasion  any  practical  inconvenience.     Sidereal  time,  as 
determined  by  the  position  of  the  true  equinox,  will  not  deviate  from  the  same  as 
indicated  by  the  position  of  the  mean  equinox,  more  than  2.3s.  in  19  years. 

350.  Another  species  of  time,  called  Mean  Equinoctial  Time,  has  recently  been 
introduced  to  some  extent  into  astronomical  calculations.     Mean  equinoctial  time 
signifies  the  mean  time  elapsed  since  the  instant  of  the  Mean  Vernal  Equinox.    Its 
use  is  to  afford  a  uniform  date,  which  shall  be  independent  of  the  different  me- 
ridians, and  of  all  inequalities  in  the  sun's  motion,  and  shall  thus  save  the  neces- 
sity, when  speaking  of  the  time  of  any  event's  happening,  of  mentioning  at  the 
same  time  the  place  where  it  was  observed  or  computed.     Thus,  it  is  the  same 
thing  to  say  that  a  comet  passed  its  perihelion  on  January  5th,  1837,  at  5h.  47m. 
0.0s.,  mean  time  at  Greenwich  ;  at  5h.  5Gm.  21.5s.,  mean  time  at  Paris  ;  or  at 
1836y.  289d.  6h.  16rn.  40.96s.,  equinoctial  time ;  but  the  former  dates  make  the 
localities  of  Greenwich  and  Paris  enter  as  elements  of  the  expression ;  whereas 
the  latter  expresses  the  period  elapsed  since  an  epoch  common  to  all  the  world, 
and  identifiable  independently  of  all  localities.     By  this  means,  all  ambiguities  in 
the  reckoning  of  time  are  supposed  to  be  avoided.* 

CONVERSION  OF  ONE  SPECIES  OF  TIME  INTO  ANOTHER. 

351.  The  difference  between  the  apparent  and  mean  time  is 
called  the  Equation  of  Time.    The  equation  of  time,  when  known, 
serves  for  the  conversion  of  mean  time  into  apparent,  and  the 
reverse. 

352.  To  find  the  equation  of  time. — The  hour  angle  of  the  sun 

*  (Nautical  Almanac  for  1837,  p.  515.) 


CONVERSION  OF  APPARENT  INTO  MEAN  SOLAR  TIME.          129 


(p.  15,  def.  16)  varies  at  the  rate 
of  360°  in  a  solar  day,  or  15°  per 
solar  hour.  If,  therefore,  its  value 
at  any  moment  be  divided  by  15, 
the  quotient  will  be  the  apparent 
time  at  that  moment.  In  like  man- 
ner, the  hour  angle  of  the  mean 
sun,  divided  by  15,  gives  the  mean 
time.  Now,  let  the  circle  VSD 
(Fig.  58)  represent  the  equator,  V 
the  vernal  equinox,  M  the  point  of 
the  equator  which  is  on  the  me- 
ridian, and  VS  the  right  ascension 
of  the  sun,  and  we  shall  have 

MS 

appar.  time  = 


Fig.  58. 


VM  — VS 


15  15 

Again,  if  we  suppose  S'  to  be  the  position  of  the  mean  sun, 
(VS'  being  equal  to  the  mean  longitude  of  the  sun,)  we  shall  have 

MS'      VM  — VS' 
mean  time  = 


15 


15- 


thus,  equa.  of  time  =  mean  time  —  ap.  time  = 


VS  — VS' 
15 


..(74); 


or,  the  equation  of  time  is  equal  to  the  difference  betiveen  the 
surfs  true  right  ascension  and  mean  longitude,  converted  into 
time. 

This  rule  will  require  some  modification  if  very  great  accuracy  is  desired ;  for, 
in  seeking  an  expression  for  the  mean  time,  the  circle  VSD  ought  properly  to  be 
considered  as  the  mean  equator,  answering  to  the  mean  pole,  (147),  and  the  mean 
longitude  of  the  sun  is  really  estimated  from  the  mean  equinox  V,  and  ought  there- 
fore to  be  corrected  by  the  arc  W,  or  the  equation  of  the  equinoxes  in  right  as- 
cension, (147.) 

The  value  of  the  equation  of  time,  determined  from  formula 
(74),  is  to  be  applied  with  its  sign  to  the  apparent  time  to  obtain 
the  mean,  and  with  the  opposite  sign  to  the  mean  time  to  obtain 
the  apparent. 

A  formula  has  been  investigated,  and  reduced  to  a  table,  which 
makes  known  the  equation  of  time  by  means  of  the  sun's  mean 
longitude.  (See  Table  XII.)  The  value  of  the  equation  of  time 
at  noon,  on  any  day  of  the  year,  is  also  to  be  found  in  the  epheni- 
eris  of  the  sun,  published  in  the  Nautical  Almanac  and  othei 
works.  If  its  value  for  any  other  time  than  noon  be  desired,  il 
may  be  obtained  by  simple  proportion. 

353.  The  equation  of  time  is  zero,  or  mean  and  true  time  are 
the  same  four  times  in  the  year,  viz.,  about  the  15th  of  April, 
the  15th  of  June,  the  1st  of  September,  and  the  24th  of  Decem- 
ber. Its  greatest  additive  value  (to  apparent  time)  is  about  14J 
minutes,  and  occurs  about  the  llth  of  February;  and  its  greatest 

17 


130  MEASUREMENT  OF  TIME. 

subtractive  value  is  about  16£  minutes,  and  occurs  about  the  3d 
of  November. 

354.  To  convert  sidereal  time  into  mean  time,  and  vice  versa. — Making  use 
of  Fig.  58  already  employed,  the  arc  VM,  called  the  Right  Ascension  of  Mid- 
Heaven,  expressed  in  time,  is  the  sidereal  time ;  VS'  is  the  right  ascension  of  the 
mean  sun,  estimated  from  the  true  equinox,  or  the  mean  longitude  of  the  sun  cor- 
rected for  the  equation  of  the  equinoxes  in  right  ascension,  (352 ;)  and  MS'  ex- 
pressed in  time,  is  the  mean  time.     Let  the  arcs  VM,  MS',  and  VS',  converted 
into  time,  be  denoted  respectively  by  S,  M,  and  L.     Now, 

VM  =  MS'  -f-  VS' ; 
or,  S  =  M  +  L..  (75);  and  M  =  S  — L..  (76). 

If  M  -}-  L  in  equation  (75)  exceeds  24  hours,  24  hours  must  be  subtracted ;  and 
if  L  exceeds  S  in  equation  (76),  24  hours  must  be  added  to  S,  to  render  the  sub- 
traction possible. 

This  problem  may  in  practice  be  solved  most  easily  by  means  f^f  an  ephemeris 
of  the  sun,  which  gives  the  value  of  S,  or  the  sidereal  time,  at  the  instant  of  mean 
noon  of  each  day,  together  with  a  table  of  the  acceleration  of  sidereal  on  mean 
solar  time,  and  the  corresponding  table  of  the  retardation  of  mean  on  sidereal  time. 

355.  The  conversion  of  apparent  time  into  sidereal,  or  sidereal  time  into  appa- 
rent, may  be  effected  by  first  obtaining  the  mean  time,  and  then  converting  this 
into  sidereal  or  apparent  time,  as  the  case  may  be. 

DETERMINATION  OF  THE  TIME  AND  REGULATION  OF  CLOCKS 
BY  ASTRONOMICAL  OBSERVATIONS. 

356.  The  regulation  of  a  clock  consists  in  finding  its  error  and 
its  rate. 

357.  The  error  of  a  mean  solar  clock  is  most  conveniently  de- 
termined from  observations  with  a  transit  instrument  of  the  time, 
as  given  by  the  clock,  of  the  meridian  passage  of  the  sun's  centre. 
The  time  noted  will  be  the  clock  time  at  apparent  noon,  and  the 
exact  mean  time  at  apparent  noon  may  be  obtained  by  applying  to 
the  apparent  time  (24h.,  or  Oh.  Om.  Os.)  the  equation  of  time  with 
its  proper  sign,  which  may  for  this  purpose  be  taken  from  the 
Nautical  Almanac  by  simple  inspection.     A  comparison  of  the 
clock  time  with  the  exact  mean  time,  will  give  the  error  of  the 
clock. 

358.  The  daily  rate  of  a  mean  solar  clock  may  be  ascertained 
by  finding  as  above  the  error  at  two  successive  apparent  noons. 
It  the  two  errors  are  the  same  and  lie  the  same  way,  the  clock  goes 
accurately  to  mean  solar  time ;  if  they  are  different,  their  differ- 
ence or  sum,  according  as  they  lie  the  same  or  opposite  ways,  will 
be  the  daily  gain  or  loss,  as  the  case  may  be. 

359.  To  find  the  error  of  a  sidereal  clock,  compute  the  true 
right  ascension  of  some  one  of  the  fixed  stars,  (see  Prob.  XXI,) 
and  note  the  time  of  its  transit ;  the  difference  between  the  time 
observed  and  the  right  ascension  in  time  will  be  the  error.     The 
error  of  the  daily  rate  is  determined  by  observing  two  successive 
transits  of  the  same  star.     The  variation  of  the  time  of  the  second 
transit  from  that  of  the  first  will  be  the  error  in  question. 

The  error  and  rate  may  be  determined  more  accurately  from 
observations  upon  several  stars,  taking  a  mean  of  the  individual 


DETERMINATION  OF  TIME. 


131 


Fig.  59. 


results.     Stars  at  a  distance  from  the  pole  are  to  be  selected,  foi 
reasons  which  have  been  already  assigned,  (58). 

360.  In  default  of  a  transit  instrument,  the  time  may  be  obtain- 
ed and  time-keepers  regulated  by  observations  made  out  of  the 
meridian.  There  are  two  methods  by  which  this  may  be  accom- 
plished, called,  respectively,  the  method  of  Single  Altitudes,  and 
the  method  of  Double  Altitudes,  or  of  Equal  Altitudes.  These 
we  will  now  explain. 

(1.)  To  determine  the  time  from  a  measured  altitude  of  the  sun, 
or  of  a  star,  its  declination  and  also  the  latitude  of  the  place  being 
given. 

Let  us  first  suppose  that  the  altitude  of  the  sun  is  taken;  cor- 
rect the  measured  altitude  for  re- 
fraction and  parallax,  and  also,  if 
the  sextant  is  the  instrument  used, 
for  the  semi-diameter  of  the  sun. 
Then,  if  Z  (Fig.  59)  represents  the 
zenith,  P  the  elevated  pole,  and  S 
the  sun  ;  in  the  triangle  ZPS  we 
shall  know  ZP  =  co-latitude,  PS 
=  co-declination,  and  ZS  =  co-alti- 
tude, from  which  we  may  compute 
the  angle  ZPS  (==  P),  which  is  the 
angular  distance  of  the  sun  from  the  meridian,  or,  if  expressed  in 
time,  the  time  of  the  observation  from  apparent  noon,  by  the  fol- 
lowing equations,  (App.,  Resolution  of  oblique-angled  spherical 
triangles,  Case  1,) 

2  k  =  ZP  +  PS  +  ZS=  co-lat.  +  co-dec.  +  co-alt.    .  .  .  (77); 

sin-PS) 

' 


sm  ZP  sin  PS 


or, 


in2  ip  =  sin  (k  —  co-lat.)  sin  (ft-  co-dec.) 
sin  (co-lat.)  sin  (co-dec.) 


sin- 


The  value  of  P  being  derived  from  these  equations  and  convert- 
ed into  time,  (see  Prob.  Ill,)  the  result  will  be  the  apparent  time 
at  the  instant  of  the  observation,  if  it  was  made  in  the  afternoon ; 
if  not,  what  remains  after  subtracting  it  from  24  hours  will  be  the 
apparent  time.  The  apparent  time  being  found,  the  mean  time 
may  be  deduced  from  it  by  applying  the  equation  of  time. 

A  more  accurate  result  will  be  obtained  if  several  altitudes  be  measured,  the  time 
of  each  measurement  noted,  and  the  mean  of  all  the  altitudes  taken  and  regarded 
as  corresponding  to  the  mean  of  the  times.  The  correspondence  will  be  sufficiently 
exact  if  the  measurements  be  all  made  within  the  space  of  10  or  12  minutes,  and 
when  the  sun  is  near  the  prime  vertical.  If  an  even  number  of  altitudes  be  taken, 
and  alternately  of  the  upper  and  lower  limb,  the  mean  of  the  whole  will  give  the 
altitude  of  the  sun's  centre,  without  it  being  necessary  to  know  his  apparent  semi- 
diameter.  In  practice,  the  declination  of  the  sun  may  be  taken  for  the  solution  of 
this  problem  from  an  ephemeris  of  the  sua.  For  this  purpose  the  time  of  the  ob» 
lervation  and  the  longitude  of  the  place  must  be  approximately  known. 


132 


MEASUREMENT  OF  TIME. 


Example.  On  the  1st  of  June,  1838,  at  about  lOh.  45m.  A.  M. 
the  altitude  of  the  sun's  lower  limb  was  measured  at  New  York 
with  a  sextant,  and  found  to  be  64°  55'  5".    What  was  the  correct 
time  of  the  observation  ?  ' 

Measured  alt.  of  the  sun's  lower  limb,     .       64°  55'    5" 
Sun's  semi-diam.,  by  Conn,  des  Terns,   .  15  47 


Appar.  alt.  of  sun's  centre,      ';'.?"     . 
Parallax  in  alt.,  (Table  X),     ., 
Refraction,  (Table  VIII),       ;>      . 

True  alt.  of  sun's  centre, 
•*>•* 

N.  York  approx.  time  of  observation, 
Diff.  of  long,  of  Paris  and  N.  York, 

Paris  approx.  time  of  obs.,  ''•         •       ".'• 

Sun's  declin.  June  1st,  M.  noon  at  Paris, 
"         "       June  2d, 


65 


10    52 

+  4 

—27 


65     10    29 

lOh.  45m. 
5       5 

3     50  P. 

22°    2'  27" 
22    10  31 

8     4 


Change  of  declin.  in  24  hours, 

24h.:8'4":  :  3h.  50m.:  1' 17". 
Declin.  June  1st,  M.  noon  at  Paris,  22°  2'  27" 


Change  of  declin.  in  3h.  50m., 
Declin  at  time  of  obs., 

90°     0'    0" 
Lat.  of  N.  York,  40    42  40 


1  17 


Co-lat.  . 
Co-dec.  . 
Co-alt.  . 

k  . 
k  —  co-lat. 
k  —  co-dec. 

.  49 
.  67 
.  24 

17  20 
56  16 
49  31 

2)142 

3  7 

.  71 
.  21 
.   3 

1  33 
44  13 
5  17 

22  3  44 


ar.  co.  sin.  0.12033 
ar.  co.  sin.  0.03303 


sin.  9.56861 
sin.  8.73135 


JP=  9  42  7.5. 
P  =  19  24  15 
4 

Ih.  17m.  37s.  0'" 

10  42      23  A.  M. 
Equa.  of  time,        —  2      34 

M.  time  of  obs.    10  39    49  A.  M. 


2)18.45332 
9.22666 


THE  CALENDAR.  133 

In  case  the  altitude  of  a  star  is  taken,  the  value  of  P  derived  from  formula  (79), 
when  converted  into  time,  will  express  the  distance  in  time  of  the  star  from  the 
meridian,  and  being  added  to  the  right  ascension  of  the  star,  if  the  observation  be 
made  to  the  westward  of  the  meridian,  or  subtracted  from  the  right  ascension  (in- 
creased  by  24h.,  if  necessary)  if  the  observation  be  made  to  the  eastward,  will  give 
the  sidereal  time  of  the  observation. 

(2.)  To  determine  the  time  of  noon  from  equal  altitudes  of  the 
sun,  the  times  of  the  observations  being  given. 

If  the  sun's  declination  did  not  change  while  he  is  above  the  hori- 
zon, he  would  have  equal  altitudes  at  equal  times  before  and  after 
apparent  noon.  Hence,  if  to  the  time  of  the  first  observation  one 
half  the  interval  of  time  between  the  two  observations  should  be 
added,  the  result  would  be  the  time  of  noon,  as  shown  by  the  clock 
or  watch  employed  to  note  the  times  of  the  observations.  The 
deviation  from  12  o'clock  would  be  the  error  of  the  clock  with  re- 
spect to  apparent  time.  The  difference  between  this  error  and  the 
equation  of  time  would  be  the  error  of  the  clock  with  respect  to 
mean  time. 

But,  as  in  point  of  fact  the  sun's  declination  is  continually  chang- 
ing, equal  altitudes  will  not  have  place  precisely  at  equal  times  be- 
fore and  after  noon,  and  it  is  therefore  necessary,  in  order  to  obtain 
an  exact  result,  to  apply  a  correction  to  the  time  thus  obtained 
This  correction  is  called  the  Equation  of  Equal  Altitudes.  Tables 
have  been  constructed  by  the  aid  of  which  the  equation  is  easily 
obtained.  This  is  at  the  same  time  a  very  simple  and  very  accu- 
rate method  of  finding  the  time  and  the  error  of  a  clock. 

If  equal  altitudes  of  a  star  should  be  observed,  it  is  evident  that 
half  the  interval  of  time  elapsed  wrould  give  the  time  of  the  star 
passing  the  meridian,  without  any  correction.  From  this  the  error 
of  the  clock  (if  keeping  sidereal  time)  may  be  found,  as  explained 
in  Art.  359. 

OF  THE  CALENDAR. 

361.  The  apparent  motions  of  the  sun,  which  bring  about  the 
regular  succession  of  day  and  night  and  the  vicissitude  of  the  sea- 
sons, and  the  motion  of  the  moon  to  and  from  the  sun  in  the  heav- 
ens, attended  with  conspicuous  and  regularly  recurring  changes  in 
her  disc,  furnish  three  natural  periods  for  the  measurement  of  the 
lapse  of  time,  viz.  1,  the  period  of  the  apparent  revolution  of  the 
sun  with  respect  to  the  meridian,  comprising  the  two  natural  pe- 
riods of  day  and  night,  which  is  called  the  solar  day  ;  2,  the  period 
of  the  apparent  revolution  of  the  sun  with  respect  to  the  equator, 
comprehending  the  four  seasons,  which  is  called  the  tropical  year ; 
3,  the  period  of  time  in  which  the  moon  passes  through  all  her 
phases  and  returns  to  the  same  position  relative  to  the  sun,  called 
a  lunar  month.  The  day  is  arbitrarily  divided  into  twenty-four 
equal  parts  called  hours ;  the  hours  into  sixty  equal  parts  called 
minutes ;  and  the  minutes  into  sixty  equal  parts  called  seconds. 


134  MEASUREMENT  OF  TIME. 

The  tropical  year  contains  365d.  5h.  48m.  48s.  The  lunar  month 
consists  of  about  29|  days.  The  week,  consisting  of  seven  days, 
has  its  origin  in  Divine  appointment  alone.  A  Calendar  is  a  scheme 
for  taking  note  of  the  lapse  of  time,  and  fixing  the  dates  of  occur- 
rences, by  means  of  the  four  periods  just  specified,  viz.  the  day, 
the  week,  the  month,  and  the  year,  or  periods  taken  as  nearly  equal 
to  these  as  circumstances  will  admit.  Different  nations  have,  in 
general,  had  calendars  more  or  less  different :  and  the  proper  ad- 
justment or  regulation  of  the  calendar  by  astronomical  observa- 
tions has  in  all  ages  and  with  all  nations  been  an  object  of  the 
highest  importance.  We  propose,  in  what  follows,  to  explain  only 
the  Julian  and  Gregorian  Calendars. 

362.  The  Julian  calendar  divides  the  year  into  12  months,  con- 
taining in  all  365  days.  Now,  it  is  desirable  that  the  calendar 
should  always  denote  the  same  parts  of  the  same  season  by  the 
same  days  of  the  same  months  :  that,  for  instance,  the  summer  and 
winter  solstices,  if  once  happening  on  the  21st  of  June  and  21st 
of  December,  should  ever  after  be  reckoned  to  happen  on  the  same 
days ;  that  the  date  of  the  sun's  entering  the  equinox,  the  natural 
commencement  of  spring,  should,  if  once,  be  always  on  the  20th 
of  March.  For  thus  the  labors  of  agriculture,  which  really  depend 
on  the  situation  of  the  sun  in  the  heavens,  would  be  simply  and 
truly  regulated  by  the  calendar. 

This  would  happen,  if  the  civil  year  of  365  days  were  equal  to 
the  astronomical ;  but  the  latter  is  greater ;  therefore,  if  the  cal- 
endar should  invariably  distribute  the  year  into  365  days,  it  would 
fall  into  this  kind  of  confusion,  that  in  process  of  time,  and  suc- 
cessively, the  vernal  equinox  would  happen  on  every  day  of  the 
civil  year.  Let  us  examine  this  more  nearly. 

Suppose  the  excess  of  the  astronomical  year  above  the  civil  to 
be  exactly  6  hours,  and  on  the  noon  of  March  20th  of  a  certain 
year,  the  sun  to  be  in  the  equinoctial  point ;  then,  after  the  lapse 
of  a  civil  year  of  365  days,  the  sun  would  be  on  the  meridian,  but 
not  in  the  equinoctial  point ;  it  would  be  to  the  west  of  that  point, 
and  would  have  to  move  6  hours  in  order  to  reach  it,  and  to  com- 
plete the  astronomical  or  tropical  year.  At  the  completions  of  a 
second  and  a  third  civil  year,  the  sun  would  be  still  more  and  more 
remote  from  the  equinoctial  point,  and  would  be  obliged  to  move, 
respectively,  for  12  and  18  hours  before  he  could  rejoin  it  and  com- 
plete the  astronomical  year. 

At  the  completion  ol  a  fourth  civil  year  the  sun  would  be  more 
distant  than  on  the  two  preceding  ones  from  the  equinoctial  point. 
In  order  to  rejoin  it,  and  to  complete  the  astronomical  year,  he 
must  move  for  24  hours ;  that  is,  for  one  whole  day.  In  other 
words,  the  astronomical  year  would  not  be  completed  till  the  be- 
ginning of  the  next  astronomical  day ;  till,  in  civil  reckoning,  the 
noon  of  March  21  st. 

At  the  end  of  four  more  common  civil  years,  the  sun  would  be 


THE    CALENDAR.  135 

in  the  equinox  on  the  noon  of  March  22d.  At  the  end  of  8  and 
64  years,  on  March  23d  and  April  6th,  respectively ;  at  the  end 
of  736  years,  the  sun  would  be  in  the  vernal  equinox  on  Septem- 
ber 20th  ;  and  in  a  period  of  about  1508  years,  the  sun  would 
have  been  in  every  sign  of  the  zodiac  on  the  same  day  of  the  cal- 
endar, and  in  the  same  sign  on  every  day. 

363.  If  the  excess  of  the  astronomical  above  the  civil  year  were 
really  what  we  have  supposed  it  to  be,  6  hours,  this  confusion  of 
the  calendar  might  be  most  easily  avoided.     It  would  be  necessa- 
ry merely  to  make  every  fourth  civil  year  to  consist  of  366  days  ; 
and,  for  that  purpose,  to  interpose,  or  to  intercalate,  a  day  in  a 
month  previous  to  March.   By  this  intercalation,  what  would  have 
been  March  21st  is  called  March  20th,  and  accordingly  the  sun 
would  be  still  in  the  equinox  on  the  same  day  of  the  month. 

This  mode  of  correcting  the  calendar  was  adopted  by  Julius 
Caesar.  The  fourth  year  into  which  the  intercalary  day  is  intro- 
duced was  called  Bissextile ;  it  is  now  frequently  called  the  Leap 
year.  The  correction  is  called  the  Julian  correction,  and  the 
length  of  a  mean  Julian  year  is  365d.  6h. 

By  the  Julian  Calendar,  every  year  that  is  divisible  by  4  is  a 
leap  year,  and  the  rest  common  years. 

364.  The  astronomical  year  being  equal  to  365d.  5h.  48m.  47.6s., 
it  is  less  than  the  mean  Julian  by  llm.  12.4s.  or  0.007782d.  The 
Julian  correction,  therefore,  itself  needs  correction.    The  calendar 
regulated  by  it  would,  in  process  of  time,  become  erroneous,  and 
would  require  reformation. 

The  intercalation  of  the  Julian  correction  being  too  great,  its 
effect  would  be  to  antedate  the  happening  of  the  equinox.  Thus 
(to  return  to  the  old  illustration)  the  sun,  at  the  completion  of  the 
fourth  civil  year,  now  the  Bissextile,  would  have  passed  the  equi- 
noctial point  by  a  time  equal  to  four  times  0.007782d. ;  at  the  end 
of  the  next  Bissextile,  by  eight  times  0.007782d. ;  at  the  end  of 
1 30  years,  by  about  one  day.  In  other  words,  the  sun  would 
have  been  in  the  equinoctial  point  24  hours  previously,  or  on  the 
noon  of  March  19th. 

In  the  lapse  of  ages  this  error  would  continue  and  be  increased. 
Its  accumulation  in  1300  years  would  amount  to  10  days,  and  then 
the  vernal  equinox  would  be  reckoned  to  happen  on  March  10th 

365.  The  error  into  which  the  calendar  had  fallen,  and  would 
continue  to  fall,  was  noticed  by  Pope  Gregory  XIII.  in  1582.   At 
his  time  the  length  of  the  yearwas  known  to  greater  precision  than 
at  the  time  of  Julius  Caesar.     It  was  supposed  equal  to  365d.  5h. 
49m.  16.23s.     Gregory,  desirous  that  the  vernal  equinox  should 
be  reckoned  on  or  near  March  21st,  (on  which  day  it  happened  in 
the  year  325,  when  the  Council  of  Nice  was  held,)  ordered  that 
the  day  succeeding  the  4th  of  October,  1582,  instead  of  being 
called  the  5th,  should  be  called  the  15th:  thus  suppressing  10 
days,  which,  in  the  interval  between  the  years  325  and  1582, 


136  MEASUREMENT  OF  TIME. 

represented  nearly  the  accumulation  of  error  arising  from  the  ex- 
cessive intercalation  of  the  Julian  correction. 

This  act  reformed  the  calendar.  In  order  to  correct  it  in  future 
ages,  it  was  prescribed  that,  at  certain  convenient  periods,  the  in- 
tercalary day  of  the  Julian  correction  should  be  omitted.  Thus 
the  centurial  years  1700,  1800,  1900,  are,  according  to  the  Julian 
calendar,  Bissextiles,  but  on  these  it  was  ordered  that  the  interca- 
lary day  should  not  be  inserted ;  inserted  again  in  2000,  but  not 
inserted  in  2100,  2200,  2300  ;  and  so  on  for  succeeding  centuries. 
By  the  Gregorian  calendar,  then,  every  centurial  year  that  is  di- 
visible by  400  is  a  Bissextile  or  Leap  year,  and  the  others  common 
years.  For  other  than  centurial  years,  the  rule  is  the  same  as  with 
the  Julian  calendar. 

366.  This  is  a  most  simple  mode  of  regulating  the  calendar.   It 
corrects  the  insufficiency  of  the  Julian  correction,  by  omitting,  in 
the  space  of  400  years,  3  intercalary  days.     And  it  is  easy  to  esti- 
mate the  degree  of  its  accuracy.     For  the  real  error  of  the  Julian 
correction  is  0.007782d.  in  1  year,  consequently  400  x  0.007782d. 
or  3.1128d.  in  400  years.     Consequently,  0.1128d.  or  2h.  42m. 
26s.  in  400  years,  or  1  day  in  3546  years,  is  the  measure  of  the 
degree  of  inaccuracy  in  the  Gregorian  correction. 

367.  The  Gregorian  calendar  was  adopted  immediately  on  its 
promulgation,  in  all  Catholic  countries,  but  in  those  where  the 
Protestant  religion  prevailed,  it  did  not  obtain  a  place  till  some 
time  after.     In  England,  "  the-  change  of  style,"  as  it  was  called, 
took  place  after  the  2d  of  September,  1752,  eleven  nominal  days 
being  then  struck  out ;  so  that  the  last  day  of  Old  Style  being  the 
2d,  the  first  of  New  Style  (the  next  day)  was  called  the  14th,  in- 
stead of  the  3d.     The  same  legislative  enactment  which  estab- 
lished the  Gregorian  calendar  in  England,  changed  the  time  of  the 
beginning  of  the  year  from  the  25th  of  March  to  the  1st  of  January. 
Thus  the  year  1752,  which  by  the  old  reckoning  would  have  com- 
menced with  the  25th  of  March,  was  made  to  begin  with  the  1st 
of  January :  so  that  the  number  of  the  year  is,  for  dates  falling 
between  the  1st  of  January  and  the  25th  of  March,  one  greater  by 
the  new  than  by  the  old  style.     In  consequence  of  the  intercalary 
day  omitted  in  the  year  1 800,  there  is  now,  for  all  dates,  1 2  days 
difference  between  the  old  and  new  style. 

Russia  is  at  present  the  only  Christian  country  in  which  the 
Gregorian  calendar  is  not  used. 

368.  The  calendar  months  consist,  each  of  them,  of  30  or  31 
days,  except  the  second  month,  Februaiy,  which,  in  a  common 
year,  contains  28  days,  and  in  a  Bissextile,  29  days  ;  the  interca- 
lary day  being  added  at  the  last  of  this  month. 

369.  To  find  the  number  of  days  comprised  in  any  number  of 
civil  years,  multiply  365  by  the  number  of  years,  and  add  to  the 
product  as  many  days  as  there  are  Bissextile  years  in  the  period. 


PART    II. 


^a   iA- 

ON  THE  PHENOMENA  RESULTING  FROM  THE  MOTIONS  OF  THE 

HEAVENLY  BODIES,  AND  ON  THEIR  APPEARANCES,  DIMEN- 
SIONS, AND  PHYSICAL  CONSTITUTION, 


CHAPTER  XIII. 

OP  THE  SUN  AND  THE  PHENOMENA  ATTENDING  ITS  APPARENT 
MOTIONS. 

INEQUALITY  OF  DAYS  * 

370.  WE  will  first  give  a.  detailed  description  of  the  sun's  ap- 
parent motion  with  respect  to  the  equator,  the  phenomenon  upon 
which  the  inequality  of  days  (as  well  as  the  vicissitude  of  the 
seasons,  soon  to  be  treated  of)  immediately  depends. 


Fig.  60. 


Let  VEAQ  (Fig.  60)  represent 
the  equator,  VT AW  (inclined  to 
VEAQ,  under  the  angle  TOE, 
measured  by  the  arc  TE,  equal 
to  23^°,)  the  ecliptic,  TnZ  and 
Wn'Z'  the  two  tropics,  POP'  the 
axis  of  the  heavens,  and  PEP'Q 
the  meridian  and  HVRA  the  ho-  B 
rizon  in  one  of  their  various  po- 
sitions with  respect  to  the  other 
circles.  About  the  21st  of  March 
the  sun  is  in  the  vernal  equinox 
V,  crossing  the  equator  in  the 
oblique  direction  VS,  towards  the 
north  and  east.  At  this  time  its  diurnal  circle  is  identical  with  the 
equator,  and  it  crosses  the  meridian  at  the  point  E,  south  of  the 
zenith  a  distance  ZE  equal  to  the  latitude  of  the  place.  Ad- 
vancing towards  the  east  and  north,  it  takes  up  the  successive 
positions  S,  S',  S",  &c.,  and  from  day  to  day  crosses  the  meridian 
at  r,  r',  &c.,  farther  and  farther  to  the  north.  Its  diurnal  circles 
will  be,  respectively,  the  northern  parallels  of  declination  passing 
through  S,  S',  S",  &c.,  and  continually  more  and  more  distant 
from  the  equator.  The  distance  of  the  sun  and  of  its  diurnal  circle 
from  the  equator,  continues  to  increase  until  about  the  2 1st  of 
June,  when  he  reaches  the  summer  solstice  T.  At  this  point  he 


*  The  day,  here  considered,  is  the  interval  between  sunrise  and  sunset. 

18 


138  OF  THE  SUN  AND  ITS  PHENOMENA. 

moves  for  a  short  time  parallel  to  the  equator :  his  declinatior 
changes  but  slightly  for  several  days,  and  he  crosses  the  meridian 
from  day  to  day  at  nearly  the  same  place.  It  is  on  this  account, 
viz.,  because  the  sun  seems  to  stand  still  for  a  time  with  respect  to 
the  equator,  when  at  the  point  90°  distant  from  the  equinox,  that 
this  point  has  received  the  name  of  solstice.*  The  diurnal  circle 
described  by  the  sun  is  now  identical  with  the  tropic  of  Cancer, 
TraZ,  which  circle  is  so  called  because  it  passes  through  T  the 
beginning  of  the  sign  Cancer,  and  when  the  sun  reaches  it,  he  is 
at  his  northern  goal,  and  turns  about  and  goes  towards  the  south.! 
The  sun  is,  also,  when  at  the  summer  solstice,  at  its  point  of  near- 
est approach  to  the  zenith  of  every  place  whose  latitude  ZE  ex- 
ceeds the  obliquity  of  the  ecliptic  TE,  equal  to  23£°.  The  distance 
ZT  =  ZE  —  ET  =  latitude  —  obliquity  of  ecliptic.  During  the 
three  months  following  the  21st  of  June,  the  sun  moves  over  the 
arc  TA,  crossing  the  meridian  from  day  to  day  at  the  successive 
points  r",  rf,  &c.,  farther  and  farther  to  the  south,  and  arrives  at 
the  autumnal  equinox  A  about  the  23d  of  September,  when  its 
diurnal  circle  again  becomes  identical  with  the  equator.  It  crosses 
the  equator  obliquely  towards  the  east  and  south,  and  during  the 
next  six  months  has  the  same  motion  on  the  south  of  the  equator, 
that  it  has  had  during  the  previous  six  months  on  the  north  of 
the  equator.  It  employs  three  months  in  passing  over  the  arc 
AW,  during  which  period  it  crosses  the  meridian  each  day  at  a 
point  farther  to  the  south  than  on  the  preceding  day.  At  the 
winter  solstice,  which  occurs  about  the  22d  of  December,  it  is 
again  moving  parallel  to  the  equator,  and  its  diurnal  circle  is  the 
same  circle  as  the  tropic  of  Capricorn.  In  three  months  more  it 
passes  over  the  arc  WV,  crossing  the  meridian  at  the  points  s",  s', 
<fec.,  so  that  on  the  21st  of  March  it  is  again  at  the  vernal  equinox. 
371.  To  explain  now  the  phenomenon  of  the  inequality  of  days 
which  obtains  at  all  places  north  or  south  of  the  equator.  At  all 
such  places,  the  observer  is  in  an  oblique  sphere ;  that  is,  the  ce- 
lestial equator  and  the  parallels  of  declination  are  oblique  to  the 
horizon.  This  position  of  the  sphere  is  represented  in  Fig.  1 1 , 
p.  21,  where  HOR  is  the  horizon,  QOE  the  equator,  and  ncr,  set, 
&c.,  parallels  of  declination  ;  WOT  is  the  ecliptic.  It  is  also  rep- 
resented in  Fig.  60,  from  which  Fig.  1 1  differs  chiefly  in  this,  that 
the  horizon,  equator,  ecliptic,  and  parallels  of  declination,  which 
are  stereographically  represented  as  ellipses  in  Fig.  60,  are  in  Fig. 
1 1  orthographically  projected  into  right  lines  upon  the  plane  of  the 
meridian.  Since  the  centres  of  the  parallels  of  declination  are 
situated  upon  the  axis  of  the  heavens,  which  is  inclined  to  the 
horizon,  it  is  plain  that  these  parallels,  as  it  is  represented  in  the 
Figs.,  and  as  we  have  before  seen,  (35,)  will  be  divided  into  un- 
equal parts,  and  that  the  disparity  between  the  parts  will  be  greater 

*  Fiom  Sol,  the  sun,  and  sto,  to  stand.  t  From  rpeirw,  to  turn. 


INEQUALITY  OF  DAYS.  139 

in  proportion  as  the  parallel  is  more  distant  from  the  equator; 
also,  that  to  the  north  of  the  equator  the  greater  parts  will  lie  above 
the  horizon,  and  to  the  south  of  the  equator  below  the  horizon. 
Now,  the  length  of  the  day  is  measured  by  the  portion  of  the 
parallel  to  the  equator,  described  by  the  sun,  which  lies  above  the 
horizon;  and  if^is  evident,  from  what  has  just  been  stated,  that 
(as  it  is  showrWby  the  Fig.)  this  increases  continually  from  the 
.winter  solstice  W  to  the  summer  solstice  T,  and  diminishes  con- 
tinually from  the  summer  solstice  T  to  the  winter  solstice  W ; 
whence  it  appears  that  the  day  will  increase  in  length  from  the 
winter  to  the  summer  solstice^  and  diminish  in  length  from  the 
summer  to  the  winter  solstice. 

372.  As  the  equator  is  bisected  by  the  horizon,  at  the  equinoxes 
the  day  and  night  must  be  each  12  hours  long. 

373.  When  the  sun  is  north  of  the  equator,  the  greater  part  of 
its  diurnal  circle  lies  above  the  horizon,  in  northern  latitudes  ;  and, 
therefore,  from  the  vernal  to  the  autumnal  equinox  the  day  is,  in 
the  northern  hemisphere,  more  than  12  hours  in  length.     On  the 
other  hand,  when  the  sun  is  south  of  the  equator,  the  greater  part 
of  its  circle  lies  below  the  horizon,  and  hence  from  the  autumnal 
to  the  vernal  equinox  the  day  is  less  than  1 2  hours  in  length. 

In  the  latter  interval  the  nights  will  obviously,  at  corresponding 
periods,  be  of  the  same  length  as  the  days  in  the  former. 

374.  The  variation  in  the  length  of  the  day  in  the  course  of  the 
year,  will  increase  with  the  latitude  of  the  place  ;  for  the  greater 
is  the  latitude,  the  more  oblique  are  the  circles  described  by  the 
sun  to  the  horizon,  and  the  greater  is  the  disparity  between  the 
parts  into  which  they  are  divided  by  the  horizon.     This  will  be 
obvious,  on  referring  to  Fig.  11,  p.  21,  where  HOR,  H'OR',  rep- 
resent the  positions  of  the  horizons  of  two  different  places  with 
respect  to  these  circles  ;  H'OR'  being  the  horizon  for  which  the 
latitude,  or  the  altitude  of  the  pole,  is  the  least. 

For  the  same  reason,  the  days  will  be  the  longer  as  we  proceed 
from  the  equator  northward,  during  the  period  that  the  sun  is 
north  of  the  equinoctial,  and  the  shorter,  during  the  period  that  he 
is  south  of  this  circle. 

375.  At  the  equator  the  horizon  bisects  ail  the  diurnal  circles, 
(36,)  and  consequently,  the  day  and  night  are  there  each  12  hours 
in  length  throughout  the  year. 

376.  At  the  arctic  circle  the  day  will  be  24  hours  long  at  the 
time  of  the  summer  solstice ;.  for,  the  polar  distance  of  the  sun 
will  then  be  66^°,  which  is  the  same  as  the  latitude  of  the  arctic 
circle  ;  whence  it  follows,  that  the  diurnal  circle  of  the  sun  at  this 
epoch,  will  correspond  to  the  circle  of  perpetual  apparition  for  the 
parallel  in  question. 

On  the  other  hand,  when  the  sun  is  at  the  winter  solstice,  the 
night  will  be  24  hours  long  on  the  arctic  circle. 

377.  To  the  north  of  the  arctic  circle,  the  sun  will  remain  con- 


140  OF  THE  SUN  AND  ITS  PHENOMENA. 

tinually  above  the  horizon  during  the  period,  before  and  after  the 
summer  solstice,  that  his  north  polar  distance  is  less  than  the  lati- 
tude of  the  place,  and  continually  below  the  horizon  during  the 
Eeriod,  about  the  winter  solstice,  that  his  south  polar  distance  is 
jss  than  the  latitude  of  the  place. 

At  the  north  pole,  as  the  horizon  is  coincident  with  the  equator, 
(37,)  the  sun  will  be  above  the  horizon  while  passing  from  the  ver- 
nal to  the  autumnal  equinox,  and  below  it  while  passing  from  the 
autumnal  to  the  vernal  equinox.  Accordingly,  at  this  locality  there 
will  be  but  one  day  and  one  night  in  the  course  of  a  year,  and  each 
will  be  of  six  months'  duration.  % 

378.  The  circumstances  of  the  duration  of  light  and  darkness 
are  obviously  the  same  in  the  southern  hemisphere  as  in  the  north- 
ern, for  corresponding  latitudes  and  corresponding  declinations  of 
the  sun. 

379.  The  latitude  of  the  place  and  the  declination  of  the  sun 
being  given,  to  find  the  times  of  the  surfs  rising  and  setting  and 
the  length  of  the  day. 

Fig.  61.    '  LetHPR(Fig.  61)  be  the  me- 

ridian, HMR  the  horizon,  and  BsD 
the  diurnal  circle  described  by  the 
sun.  The  hour  angle  EP£,  or  its 
measure  E^,  which  converted  into 
time  expresses  the  interval  between 
the  rising  or  setting  of  the  sun  and 
his  passage  over  the  meridian,  is 
called  the  Semi-diurnal  Arc.  Now, 

E*  =  EM  +  M*  =  90°  +  M£, 

which  gives 

cos  E£  =  —  sin  M£  ; 
and  we  have,  by  Napier's  first  rule, 

sin  Mt =cot  tM.s  tang  ts  --  tang  PMH  tang  EB  =tang  PH  tang  EB  : 
whence,  cos  E£  =  —  tang  PH  tang  EB, 

or,       cos  (semi-diurnal  arc)  =  -r-  tang  lat.  x  tang  dec.  .  .  (80). 

The  semi-diurnal  arc  (in  time)  expresses  the  apparent  time  of 
the  sun's  setting;  and  subtracted  from  12  hours,  gives  the  appa- 
rent time  of  its  rising.  The  double  of  it  will  be  the  length  of  the  day. 
In  resolving  this  problem  it  will,  in  practice,  generally  answer  to 
make  use  of  the  declination  of  the  sun  at  noon  of  the  given  day, 
which  may  be  taken  from  an  ephemeris. 

Exam.  1 .  Let  it  be  required  to  find  the  apparent  times  of  the 
sun's  rising  and  setting  and  the  length  of  the  day  at  New  York  at 
the  summer  solstice. 

Log.  tang  lat.  (40°  42'  40")  .         .-        . .       9.93474  — 
Log.  tang  dec.  (23°  27'  40")          .         .         9.63749 

Log.  cos  (semi-diurnal  arc)  .         .         .         9.57223 — 


TWILIGHT.  14) 

Semi-diurnal  arc        ,-,«:> .  * •••$£& •'?**$     •    HI0   55'     40" 
Time  of  sun's  setting  >,    v..-.         ...      ,?, ;  7h.  27m. 43s. 
Time  of  sun's  rising    -«  •  *.  '^..    ;  .^  .  *:      4     32      17 
Length  of  day      .       . »    •  >  <(,i  ;,<VN  5  ^  >  14     55      26 

Exam.  2.  What  are  the  lengths  of  the  longest  and  shortest  days 
at  Boston ;  the  latitude  of  that  place  being  42°  21'  15"  N.? 

Ans.  15h.  6m.  28s.  and  8h.  53m.  32s. 

Exam.  3.  At  what  hours  did  the  sun  rise  and  set  on  May  1st, 
1837,  at  Charleston;  the  latitude  of  Charleston  being  32°  47',  and 
the  declination  of  the  sun  being  15°  6'  0"  N.  ? 

Ans.  Time  of  rising,  5h.  19m.  58s.  Time  of  setting,  6h.  40m. 
2s. 

380.  To  find  the  time  of  the  surfs  apparent  rising  or  setting; 
the  latitude  of  the  place  and  the  declination  of  the  sun  being  given. 

At  the  time  of  his  apparent  rising  or  setting,  the  sun  as  seen  from 
the  centre  of  the  earth  will  be  below  the  horizon  a  distance  sS 
(Fig.  61)  equal  to  the  refraction  minus  the  parallax.  The  mean 
difference  of  these  quantities  is  33'  42".  Let  it  be  denoted  by  R. 
Now,  to  find  the  hour  angle  ZPS(=P),  the  triangle  ZPS  gives, 
(see  Appendix,) 

f       ZP  +  PS  +  ZS       co-lat.  +  co-dec.  -f  (90°-f  R) 
k  =  -       ~2~  ~2~  --..(81) 

|1D      sin  (k  -ZP)  sin  (k  -PS) 

and  siir  |r  = : — ^^   .    po , 

sin  ZP  sin  PS 

.  „   _        sin  (k  —  co-lat.)  sin  (A;  —  co-dec.)  /0rtX 

or,          sinaJP  = --r—. — r — ^—^ — -j — , '-    .  .  .  (82). 

sin  (co-lat.)  sin  (co-dec.) 

The  value  of  P  (in  time)  will  be  the  interval  between  apparent 
noon  and  the  time  of  the  apparent  rising  or  setting. 

If  the  time  of  the  rising  or  setting  of  the  upper  limb  of  the  sun, 
instead  of  its  centre,  be  required,  we  must  take  for  R  33'  42"  -f 
sun's  semi-diameter,  or  49'  43". 

Unless  very  accurate  results  are  desired,  it  will  be  sufficient  to 
take  the  declinations  of  the  sun  at  6  o'clock  in  the  morning  and 
evening.  When  the  greatest  precision  is  required,  the  times  of  true 
rising  and  setting  must  be  computed  by  equation  (80),  and  the  de- 
clinations found  for  these  times. 

TWILIGHT. 

381.  When  the  sun  has  descended  below  the  horizon,  its  rays 
still  continue  to  fall  upon  a  certain  portion  of  the  body  of  air  that 
lies  above  it,  and  are  thence  reflected  down  upon  the  earth,  so  as 
to  occasion  a  certain  degree  of  light,  which  gradually  diminishes  as 
the  sun  descends  farther  below  the  horizon,  and  the  portion  of  the 
air  posited  above  the  horizon,  that  is  directly  illuminated,  becomes 
less.     The  same  effect,  though  in  a  reverse  order,  takes  place  in 


142 


OF  TIJE  SUN  AND  ITS  PHENOMENA. 


the  morning  previous  to  the  sun's  rising.  The  light  thus  produced 
is  called  the  Crepusculum,  or  Twilight.  This  explanation  of  twi- 
light will  be  better  understood  on  examining  Fig.  62,  where  AON 
represents  a  portion  of  the  earth's  surface,  H/cR  the  surface  of  the 

Fig.  62. 


atmosphere  above  it,  and  kmS  a  line  drawn  touching  the  earth  and 
passing  through  the  sun.  The  unshaded  portion,  kcR,  of  the  body 
of  air  which  lies  above  the  plane  of  the  horizon  HOR,  is  still  illu- 
minated by  the  sun,  and  shines  down,  by  reflection,  upon  O  the 
station  of  the  observer.  As  the  sun  descends  this  will  decrease, 
until  finally  when  the  sun  is  in  the  direction  RNS'  he  will  illumi- 
nate directly  none  of  that  part  of  the  atmosphere  which  lies  above 
the  horizon,  and  twilight  will  be  at  an  end. 

382.  The  close  of  the  evening  twilight  is  marked  by  the  ap- 
pearance of  faint  stars  over  the  western  horizon,  and  the  beginning 
of  the  morning  twilight  by  the  disappearance  of  faint  stars  situated 
in  the  vicinity  of  the  eastern  horizon.    It  has  been  ascertained  from 
numerous  observations,  that,  at  the  beginning  of  the  morning  and 
end  of  the  evening  twilight,  the  sun  is  about  18°  below  the  horizon. 

383.  At  this  time,  then,  the  angle  TRS'  is  equal  to  18°.     This  datum  will  ena- 
ble us  to  calculate  the  approximate  height  of  the  atmosphere.     For  if  the  verticals 
at  O,  m,  and  N  be  produced  to  the  centre  of  the  earth,  we  shall  have  the  angle 
OCN  equal  to  TRS',  or  18°,  and  therefore  OCR  equal  to  9°  ;  and  thus  the  height 
of  the  atmosphere,  mR,  equal  to  CR  —  Cm,  equal  to  secant  of  9°  —  radius.  Making 
the  calculation,  we  find  the  height  of  the  atmosphere  to  be  about  47  miles.     It  is 
to  be  understood  that  this  is  only  a  rough  approximation. 

It  will  be  seen,  on  inspecting  Fig.  62,  that  twilight  would  continue  longer  if  the 
atmosphere  were  higher. 

384.  The  latitude  of  the  place  and  the  sun's  declination  being 
given,  to  find  the  time  of  the  beginning  or  end  of  twilight. 

The  zenith  distance  of  the  sun  at  the  beginning  of  morning  or 
end  of  evening  twilight,  is  90° -f  18°  :  wherefore  we  may  solve  this 
problem  by  means  of  equations  (81)  and  (82),  taking  R  =  18°. 

If  the  time  of  the  commencement  of  morning  twilight  be  sub- 
tracted from  the  time  of  sunrise,  the  remainder  will  be  the  dura- 
tion of  twilight. 

At  the  latitude  49°,  the  sun  at  the  time  of  the  summer  solstice 
is  only  18°  below  the  horizon,  at  midnight ;  for  the  altitude  of  the 


TWILIGHT. 


143 


pole  at  a  place  the  latitude  of  which  is  49°,  differs  only  18°  from 
the  polar  distance  of  the  sun  at  this  epoch.  This  may  be  illustra- 
ted by  Fig.  60,  taking  Z  as  the  point  of  passage  of  the  sun  across 
the  inferior  meridian,  PZ=67°,  and  PH  =49°.  At  this  latitude, 
therefore,  twilight  will 'continue  all  night,  at  the  summer  solstice. 
This  will  be  true  for  a  still  stronger  reason  at  higher  latitudes. 

385.  The  duration  of  twilight  varies  with  the  latitude  of  the 
place  and  with  the  time  of  the  year.  At  all  places  in  the  northern 
hemisphere,  the  summer  are  longer  than  the  winter  twilights  ;  and 
the  longest  twilights  take  place  at  the  summer  solstice  ;  while  the 
shortest  occur  when  the  sun  has  a  small  southern  declination,  dif- 
ferent for  each  latitude.*  The  summer  twilights  increase  in  length 
from  the  equator  northward. 

These  facts  are  consequences  of  the  different  situations  with  respect  to  the  hori- 
zon of  the  centres  6f  the  diurnal  circles  described  by  the  sun  in  the  course  of  the 
year,  and  of  the  different  sizes  of  these  circles.  To  make  this  evident,  let  us  con- 
ceive a  circle  to  be  traced  in  the  heavens  parallel  to  the  horizon,  and  at  the  dis- 
tance of  18°  below  it :  this  is  called  the  Crepusculum  Circle.  The  duration  of 
twilight  will  depend  upon  the  number  of  degrees  in  the  arc  of  the  diurnal  circle  of 
the  sun,  comprised  between  the  horizon  and  the  crepusculum  circle,  which,  for  the 
sake  of  brevity,  we  will  call  the  arc  of  twilight :  and  this  will  vary  from  the  two  causes 
just  mentioned.  For,  let  hkr  (Fig.  63)  represent  the  equator,  and  h'k'r'  a  diurnal 


Fig.  63. 


circle  described  by  the  sun  when  north 
of  the  equator  ;  and  let  hr,  st,  and  A  V, 
s't'f  be  the  intersections  of  the  equator 
and  diurnal  circle,  respectively,  with  the 
planes  of  the  horizon  and  crepusculum 
circle.  When  the  sun  is  in  the  equator, 
the  arc  of  twilight  is  hs,  and  when  he  is 
on  the  parallel  of  declination  h'k'r1  it  is 
AY.  Draw  the  chords  hs,  AY,  mn,  and 
the  radii  cs,  cs1,  cr1,  en,  cp.  The  angle  /* 
r'AY  is  the  half  of  r'cs1,  and  the  angle  f 
pmn  is  the  half  of  pen :  but  r'cs'  is  less 
than  pen,  and  therefore  r'AY  is  less  than 
pmn.  Again,  chs  is  the  half  of  res,  and 
therefore  greater  than  pmn,  the  half  of 
the  less  angle  pen.  Whence  it  appears 
that  the  chord  AY  is  more  oblique  to  the 
horizon,  and  therefore  greater  than  the 
chord  mn,  and  this  more  oblique  and  greater  than  the  chord  As.  It  follows,  there- 
fore, that  the  arc  AY  is  greater,  and  contains  a  greater  number  of  degrees  than  the 
arc  mn,  arid  that  this  arc  is  greater  than  A*.  Thus,  as  the  sun  recedes  from  the 
equator  towards  the  north,  the  arc  of  twilight,  and  therefore  the  duration  of  twilight, 
increases  from  two  causes,  viz  :  1st.  The  increase  in  the  distance  of  the  line  of  in- 
tersection of  the  horizon  with  the  diurnal  circle  from  the  centre  of  the  circle  ;  and, 
2d.  The  diminution  in  the  size  of  the  circle.  The  change  will  manifestly  be  greater 
in  proportion  as  the  latitude  is  greater. 

*  The  duration  of  shortest  twilight  is  given  by  the  following  formula : 

sin  9° 


~  coslat. 

Twice  the  angle  a,  converted  into  time,  expresses  the  duration  of  shortest  twilight 
To  find  the  sun's  declination  at  the  time  of  shortest  twilight,  we  have 

sin  dec.  =  — tang  9°  sin  lat 

(For  the  investigation  of  this  and  the  preceding  formula,  see  Gummere's  Astrono- 
my, pages  87  and  88.) 


144  OF  THE  SUN  AND  ITS  PHENOMENA. 

When  the  sun  is  south  of  the  equator  twilight  will,  for  the  same  declination, 
be  shorter  than  when  he  is  north  of  the  equator,  because,  although  the  diurnal  cir 
cle  will  be  of  the  same  size,  and  its  intersection  with  the  horizon  at  the  same  dis- 
tance from  its  centre,  the  intersection  with  the  crepusculum  circle  will  now  fall 
between  the  intersection  with  the  horizon  and  the  centre,  and  therefore,  by  what 
has  just  been  demonstrated,  the  arc  of  twilight  will  be  shorter. 

The  shortest  twilight  occurs  when  the  sun  is  somewhat  to  the  south  of  the  equa- 
tor, because  the  arc  of  twilight,  for  a  time,  decreases  by  reason  of  the  diminution 
of  its  obliquity  to  the  horizon  more  than  it  increases  in  consequence  of  the  decrease 
in  the  size  of  the  diurnal  circle.  That  the  obliquity  of  the  arc  of  twilight,  or  rather 
of  the  chord  of  the  arc,  to  the  horizon  diminishes,  for  a  time,  when  the  sun  gets  to 
the  south  of  the  equator,  will  appear  from  this,  viz.  that  the  chord  is  perpendicular 
to  the  horizon  when  the  centre  of  the  diurnal  circle  is  midway  between  the  horizon 
and  the  crepusculum  circle ;  which  will  happen  when  the  sun  is  a  certain  dis- 
tance south  of  the  equator,  varying  with  the  inclination  of  the  axis  of  the  heavens 
to  the  plane  of  the  horizon,  and  therefore  with  the  latitude  of  the  place. 

The  difference  in  the  length  of  the  summer  and  winter  twilights,  resulting  from 
the  causes  above  specified,  is  augmented  by  the  inequality  in  the  height  of  the  at- 
mosphere. Twilight  also  increases  in  length  with  the  obliquity  of  the  sphere. 

386.  At  the  poles  twilight  commences  about  a  month  and  a  half 
before  the  sun  appears   above   the  horizon,   and  lasts   about  a 
month  and  a  half  after  he  has  disappeared.     For,  since  the  hori- 
zon at  the  poles  is  identical  with  the  celestial  equator,  the  twilight 
which  precedes  the  long  day  of  six  months  will  begin  when  the  sun 
in  approaching  the  equator,  upon  the  other  side,  attains  to  a  decli- 
nation of  18°,  and  this  will  be  about  50  days  before  he  reaches  the 
equator  and  rises  at  the  pole.    In  like  manner  the  evening  twilight 
continues  until  the  sun  has  descended  18°  below  the  equator. 

THE  SEASONS. 

387.  The  amount  of  heat  received  from  the  sun  in  the  course 
of  24  hours,  depends  upon  two  particulars ;  the  time  of  the  sun's 
continuance  above  the  horizon,  and  the  obliquity  of  his  rays  at 
noon.    Ey  reason  of  the  obliquity  of  the  ecliptic,  both  of  these  cir- 
cumstances vary  materially  in  the  course  of  the  year ;  whence 
arises  a  variation  of  temperature  or  a  change  of  seasons. 

388.  The  tropics  and  the  polar  circles  divide  the  earth  into  five 
parts,  called  Zones,  throughout  each  of  which  the  yearly  change 
of  the  temperature  is  occasioned  by  a  similar  change  in  the  cir- 
cumstances upon  which  it  depends. 

The  part  contained  between  the  two  tropics  is  called  the  Torrid 
Zone;  the  two  parts  between  the  tropics  and  polar  circles  are 
called  the  Temperate  Zones ;  and  the  other  two  parts,  within  the 
polar  circles,  are  called  Frigid  Zones. 

389.  At  all  places  in  the  nprth  temperate  zone  the  sun  will  al- 
ways pass  the  meridian  to  the  south  of  the  zenith ;  for  the  latitudes 
of  all  such  places  exceed  23^°,  the  greatest  decimation  of  the  sun. 
(See  Fig.  60.)     The  meridian  zenith  distance  will  be  greatest  at 
the  winter  solstice,  when  the  sun  has  its  greatest  southern  decli- 
nation, and  least  at  the  summer  solstice,  when  the  sun  has*  its 
greatest  northern  declination ;  and  it  will  vary  continually  between 


THE    SEASONS.  145 

the  values  which  obtain  at  these  epochs.  The  day  will  be  longest 
at  the  summer  solstice,  and  the  shortest  at  the  winter  solstice,  and 
will  vary  in  length  progressively  from  the  one  date  to  the  other. 

We  infer,  therefore,  thaF  throughout  the  zone  in  question  the 
greatest  amount  of  heat  will  be  received  from  the  sun  at  the  sum- 
mer solstice,  and  the  least  at  the  winter  solstice ;  and  that  the 
amount  received  will  gradually  increase,  or  decrease,  from  one  of 
these  epochs  to  the  other.  The  solstices  are  not,  however,,the 
epochs  of  maximum  and  minimum  temperature,  but  are  found 
from  observation  to  precede  these  by  about  a  month.  The  reason 
of  this  circumstance  is,  that  the  earth  continues  for  a  month,  or 
thereabouts,  after  the  summer  solstice  to  receive  during  the  day 
more  heat  than  it  loses  during  the  night,  and  for  about  the  same 
length  of  time  after  the  winter  solstice  continues  to  lose  during  the 
night  more  heat  than  it  receives  during  the  day. 

390.  Within  the  torrid  zone  the  length  of  the  day  varies  after 
the  same  manner  as  in  the  temperate  zone,  though  in  a  less  de- 
gree ;  but  the  motion   of  the  sun  with  respect  to  the  zenith  is 
different.     At  all  places  in  the  torrid  zone  the  sun  passes  the  me- 
ridian during  a  certain  portion  of  the  year  to  the  south  of  the  zenith, 
and  during  the  remaining  portion  to  the  north  of  it ;  for  all  places 
so  situated  have  their  zeniths  between  the  tropics  in  the  heavens, 
and  the  sun  moves  from  one  tropic  to  the  other,  and  back  again  to 
its  original  position,  in  a  tropical  year.     Throughout  the  torrid 
zone,  therefore,  the  sun  will  be  in  the  zenith  twice  in  the  course  of 
the  year,  and  will  be  at  its  maximum  distance  from  it  on  the  one 
side  and  the  other  at  the  solstices. 

An  inhabitant  of  the  equator  or  its  vicinity,  will  have  summer 
at  the  two  periods  when  the  sun  is  in  the  zenith,  and  winter  (or  a 
period  of  minimum  temperature)  both  at  the  summer  and  winter 
solstice.  Near  the  tropic  there  will  be  but  little  variation  in  the 
daily  amount  of  heat  received,  during  the  period  that  the  sun  is 
north  of  the  zenith. 

391 .  At  the  frigid  zone  a  new  cause  of  a  change  of  temperature 
exists  ;  the  sun  remains  continually  above  the  horizon  for  a  greater 
or  less  number  of  days  about  the  summer  solstice,  and  continually 
below  it  for  the  same  number  of  days  about  the  winter  solstice. 

392.  The  amount  of  the  yearly  variation  of  temperature  in- 
creases with  the  latitude  of  the  place  ;  for  the  greater  is  the  lati- 
tude the  greater  will  be  the  variation  in  the  length  of  the  day. 
Also,  the  mean  yearly  temperature  is  lower  as  we  recede  from  the 
equator  and  approach  the  poles  ;  for  since  the  sun  is,  in  the  course 
of  the  year,  the  same  length  of  time  above  the  horizon,  at  all 
places,  the  mean  yearly  temperature  must  depend  altogether  upon 
the  mean  obliquity  of  the  sun's  rays  at  noon,  and  this  increases 
with  the  latitude. 

393.  The  yearly  change  in  the  sun's  distance  from  the  earth  has 
but  little  effect  in  producing  a  variation  of  temperature  upon  the 

19 


146  OF  THE  SUN  AND  ITS  PHENOMENA. 

earth's  surface.     The  change  of  its  heating  power  from  this  cause 
amounts  to  no  more  than  /y. 

394.  It  is  important  to  observe,  that,  although  in  the  main  cli- 
mate varies  with  the  latitude  after  ttt%  manner  explained  in  the 
foregoing  articles,  it  is  still  dependent  more  or  less  upon  local 
circumstances,  such  as  the  vicinity  of  lakes,  seas,  and  mountains, 
prevailing  winds  of  some  particular  direction,  &c. 

395.  In  the  north  temperate  zone,  Spring,  Summer,  Autumn, 
and  Winter,  the  four  seasons  into  which  the  year  is  divided,  are 
considered  as  respectively  commencing  at  the  times  of  the  Ver- 
nal Equinox,  Summer  Solstice,  Autumnal  Equinox,  and  Winter 
Solstice. 

Let  V  (Fig.  64)  represent  the  vernal,  and  A  the  autumnal  equi- 
nox ;  S  the  summer,  and  W  the  winter  solstice.     The  perigee  of 

Fig.  64. 


the  sun's  apparent  orbit  is  at  present  about  10°  15'  to  the  east  of 
the  winter  solstice.  Let  P  denote  its  position.  The  lengths  of 
the  seasons  are,  agreeably  to  Kepler's  law  of  areas,  respectively 
proportional  to  the  areas  VES,  SEA,  AEW,  and  WEV.  Thus, 
the  winter  is  the  shortest  season,  and  the  'summer  the  longest ; 
and  spring  is  longer  than  autumn.  Spring  and  summer,  taken 
together,  are  about  8  days  longer  than  autumn  and  winter  united. 

Since  the  perigee  of  the  sun's  orbit  has  a  progressive  motion, 
the  relative  lengths  of  the  seasons  must  be  subject  to  a  continual 
variation. 

396.  At  the  beginning  of  the  year  1800,  the  longitude  of  the 
sun's  perigee  was  279°  SO'  8",39.  If  from  this  we  take  180°,  the 
longitude  of  the  autumnal  equinox,  the  remainder,  99°  30'  8". 39, 
is  the  distance  of  the  perigee  from  the  autumnal  equinox  at  that 
epoch.  The  motion  of  the  perigee  in  longitude  is  at  the  rate  of 
61".52  per  year.  Dividing  99°  30'  8".39  by  61".52,  the  quotient 
is  5822.  Hence  it  appears  that  about  5800  years  anterior  to  the 


DIMENSIONS    OF   THE    SUN.  147 

year  1800,  the  perigee  coincided  with  the  autumnal  equinox,  and 
the  apogee  with  the  vernal  equinox. 

397.  It  is  important  to  observe  that  the  primary  cause  of  the 
phenomenon  of  change  of  seasons,  as  well  as  of  that  of  the  ine- 
quality of  days,  is  the  inclination  of  the  earth's  axis  of  rotation  to 
the  perpendicular  to  the  plane  of  its  orbit,  since  this  is  the  occa- 
sion of  the  obliquity  of  the  ecliptic,  upon  which,  as  we  have  seen, 
these  phenomena  immediately  depend.      If  the  axis  of  rotation 
were  perpendicular  to  the  plane  of  the  orbit,  there  would  neither 
be  a  change  of  seasons  nor  any  inequality  in  the  length  of  the  days 
and  nights. 

APPEARANCE,    DIMENSIONS,   AND  PHYSICAL  CONSTITUTION 

OF  THE  SUN. 

398.  The  sun  presents  the  appearance  of  a  luminous  circular 
disc.     But  it  does  not  necessarily  follow  from  this  that  its  surface 
is  really  flat ;  for  such  is  the  appearance  of  all  globular  bodies 
when  viewed  at  a  great  distance.     It  is  ascertained  from  observa- 
tions with  the  telescope,  that  the  sun  has  a  rotatory  motion  :  this  be- 
ing the  fact,  its  surface  must  in  reality  be  of  a  spherical  form  ;  for 
otherwise  it  would  not,  in  presenting  all  its  sides,  always  appear 
under  the  form  of  a  circle. 

399.  The  sun's  real  diameter  is  determined  from  his  apparent 
diameter  and  horizontal  parallax.  Fig.  65. 

Let  ACB  (Fig.  65)  represent  the 
sun  or  other  heavenly  body,  and 
E  the  place  of  the  earth ;  and 
let  5  =  AEB  the  sun's  apparent 
diameter,  d  —  2AS  his  real  di- 
ameter, D  =  ES  his  distance 
from  the  earth,  and  R  =  the  radius  of  the  earth.  We  have,  from 
the  triangle  AES, 

AS  =  ES  sin  1AEB,  or,  2AS  =  2ES  sin  JAEB  ; 
and  thus,  d  =  2D  sin  £<5 : 

but,  (equa.  7,)  D==^TH> 

whence,         i^^^O^^^fj^ 

The  mean  apparent  diameter  of  the  sun  is  32'  1".8,  and  his 
mean  horizontal  parallax  8".58.  Accordingly  we  have,  for  the 
real  diameter  of  the  sun, 

OO/   1  //    Q 

^  =  2RWr^y=2R  xi  12  (nearly.) 

Thus  the  diameter  of  the  sun  is  about  112  times  the  diameter 
of  the  earth.  The  volume  of  the  sun  then  exceeds  that  of  the 
earth  nearly  in  the  proportion  1123  to  I3,  or  1,404,928  to  I. 


148  OF  THE  SUN  AND  ITS  PHENOMENA. 

From  equation  (83)  we  may  derive  the  proportion 
d  :  2R  :  :  5  :  2H. 

Thus,  the  real  diameter  of  a  heavenly  body  is  to  the  diameter 
of  the  earthy  as  the  apparent  diameter  of  the  body  is  to  double  its 
horizontal  parallax. 

400.  When  the  sun  is  viewed  with  a  telescope  of  considerable 
power,  and  provided  with  colored  glasses,  black  spots  of  an  irreg- 
ular form,  surrounded  by  a  dark  border  of  a  nearly  uniform  shade, 

Fig.  66.  called  a  penumbra,  are  often  seen 

on  its  disc,  (see  Fig.  66.)  Some- 
times  several  spots  are  included 
i:.  within  the  same  penumbra.  Their 
number,  magnitude,  and  position 
- :  i;;:;||$f::::  on  the  disc,  are  extremely  variable. 
In  some  years  they  are  very  fre- 
quent, and  appear  in  large  numbers ; 
in  others,  none  whatever  are  seen. 
^n  some  instances  more  than  one 
hundred,  of  various  forms  and  sizes, 
have  been  counted.  They  usually 
appear  in  clusters,  composed  of  various  numbers,  from  two  to  sixty 
or  a  hundred.  Their  absolute  magnitude  is  often  very  great. 
Spots  are  not  unfrequently  seen  that  subtend  an  angle  of  1'  or  60". 
Now,  the  apparent  diameter  of  the  earth  as  viewed  at  the  distance 
of  the  sun,  is  equal  to  double  the  sun's  horizontal  parallax,  or  11"  : 
the  breadth  of  such  spots  must  therefore  exceed  three  times  the 
diameter  of  the  earth,  or  24,000  miles.  Spots  two  or  three  times 
as  large  as  this,  or  about  three  times  as  great  as  the  entire  surface 
of  our  globe,  have  been  seen. 

401.  The  form  and  size  of  the  spots  are  subject  to  rapid  and 
almost  incessant  variations.     When  watched  from  day  to  day,  or 
even  from  hour  to  hour,  they  are  seen  to  enlarge  or  contract,  and 
at  the  same  time  to  change  their  forms.     When  a  spot  disappears, 
it  always  contracts  into  a  point,  and  vanishes  before  the  penumbra. 
Some  spots  disappear  almost  immediately  after  they  become  visi- 
ble ;  others  remain  for  weeks,  or  even  months. 

402.  Spots  and  streaks  more  luminous  than  the  general  body 
of  the  sun,  and  of  a  mottled  appearance,  are  also  frequently  per- 
ceived upon  parts  of  his  disc,  especially  in  the  region  of  large 
spots,  or  of  extensive  groups  of  spots,  or  in  localities  where  dark 
spots  subsequently  make  their  appearance.     These  are  called  Fa- 
culce.     They  are  chiefly  to  be  seen  near  the  margin  of  the  disc. 
The  penumbra  which  surrounds  each  black  spot  is  also  abruptly 
terminated  by  a  border  of  light  more  brilliant  than  the  rest  of  the 
disc. 

According  to  Sir  John  Herschel,  the  part  of  the  sun's  disc  not 
occupied  b.y  spots  is  far  from  uniformly  bright.  Its  ground  is 
finely  mottled  with  an  appearance  of  minute  dark  dots,  or  pores, 


SUN'S  SPOTS;  AND  ROTATION.  149 

which,  when  attentively  watched,  arc  found  to  be  in  a  constant 
state  of  change. 

403.  When  the  positions  of  the  spots  on  the  disc  are  observed 
from  day  to  day,  it  is  perceived  that  they  all  have  a  common  mo- 
tion in  a  direction  from  east  to  west.     Some  of  the  spots  close  up 
and  vanish  before  they  reach  the  western  limb  ;  others  disappear 
at  the  western  limb,  and  are  never  afterwards  seen  ;  a  few,  after 
becoming  visible  at  the  eastern  limb,  have  been  seen  to  pass  en- 
tirely across  the  disc,  disappear  from  view  at  the  western  limb, 
and  re-appear  again  at  the  eastern  limb.     The  time  employed  by 
a  spot  in  traversing  the  sun's  disc  is  about  14  days.     About  the 
same  time  is  occupied  in  passing  from  the  western  to  the  eastern 
limb,  while  it  is  invisible.     The  motions  of  the  spots  are  account- 
ed for,  in  all  their  circumstances,  by  supposing  that  the  sun  has  a 
motion  of  rotation  from  west  to  east,  around  an  axis  nearly  per- 
pendicular to  the  plane  of  the   ecliptic  ;  and  that  the  spots  are 
portions  of  the  solid  body  of  the  sun.     The  truth  of  this  explana- 
tion of  the  apparent  motions  of  the  sun's  spots,  is  confirmed  by 
the  changes  which  are  observed  to  take  place  in  the  magnitude 
and  form  of  the  more  permanent  spots  during  their  passage  across 
the  disc.     When  they  first  come  into  view  at  the  eastern  limb, 
they  appear  as  a  narrow  dark  sfreak.     As  they  advance  towards 
the  middle  of  the  disc,  they  gradually  open  out,  and  increase  in 
magnitude ;  and  after  they  have  passed  the  middle  of  the  disc, 
contract  by  the  same  degrees  until  they  are  again  «seen  as  a  mere 
dark  line  upon  the  western  limb. 

404.  A  spot  returns  to  the  same  position  on  the  disc  in  about 
27|  days.     This  is  not,  however,  the  precise  period  of  the  sun's 
rotation ;  for  during  this  interval  the   sun  has  apparently  moved 
forward  nearly  a  sign  in  the  ecliptic  ;  the  spot  will  therefore  have 
accomplished  that  much  more  than  a  complete  revolution,  when  it 
is  again  seen  by  an  observer  on  the  earth  in  the  same  position  on 
the  disc. 

405.  The  apparent  position  of  a  spot  with  respect  to  the  sun's 
centre  may  be  accurately  determined,  from  day  to  Jay,  by  observ- 
ing, when  the  sun  is  crossing  the  meridian,  the  right  ascensions 
and  declinations  both  of  the  spot  and  centre.      From  three  or 
more  observations  of  this  kind  the  period  of  the  sun's  rotation  and 
the  position  of  his  equator  may  be  ascertained. 

The  time  of  the  sun's  rotation  on  his  axis  is  about  25|  days ; 
the  inclination  of  his  equator  to  the  ecliptic  7°  30' ;  and  the  helio 
centric  longitude  of  the  ascending  node  of  the  equator  80°  7'. 

406.  It  is  a  curious  fact,  that  the  region  of  the  sun's  spots  is  con 
fined  within  about  30°  of  his  equator.     It  is  only  occasionally  that 
spots  are  seen  in  higher  latitudes  than  this  :  and  none  are  ever  seen 
farther  than  about  60°  from  the  equator. 

407.  The  only  theories  relative  to  the  physical  constitution  of 
the  sun  which  deserve  notice>  are  those  of  Laplace  and  Herschel 


150  OF  THE  SUN  AND  ITS  PHENOMENA. 

Laplace  supposed  that  the  sun  was  an  immense  globe  of  solid  mat- 
ter in  a  state  of  ignition,  and  that  the  spots  upon  his  disc  were  large 
cavities,  where  there  was  a  temporary  intermission  in  the  evolution 
of  luminous  matter.  Sir  W.  Herschel  was  of  opinion  that  the  sun 
was  an  opake  solid  body,  surrounded  by  a  transparent  atmosphere 
of  tens  of  thousands  of  miles  in  height,  within  which  floated  at  a 
height  of  from  two  to  three  thousand  miles  above  the  solid  globe  a 
stratum  of  self-luminous  clouds,  which  was  the  source  of  the  sun's 
light  and  heat,  and  beneath  this  another  opake  and  non-luminous 
stratum,  which  shone  only  with  the  light  received  from  the  upper 
stratum.  On  this  hypothesis  the  spots  are  accounted  for  by  sup- 
posing that  openings  occasionally  take  place  in  the  strata,  through 
which  the  dark  body  of  the  sun  is  seen.  The  penumbra  is  the  por- 
tion of  the  obscure  stratum,  situated  immediately  around  the  open- 
ing made  in  it.  This  theory  seems  to  account  for  all  the  circum- 
stances of  the  aspect  and  variation  of  the  form  and  magnitude  of 
the  spots,  which  the  other  does  not  do. 

408.  That  the  dark  spots  are  depressions  below  the  luminous  surface  of  the  sun 
was  first  shown  by  Dr.  Alexander  Wilson,  of  Glasgow.  He  noticed  that  as  a  large 
spot,  which  was  seen  on  the  sun's  disc  in  November,  17 69,  came  near  the  western  limb, 
the  penumbra  on  the  side  towards  the  centre  of  the  disc  contracted  and  disappeared, 
and  that  afterwards  the  luminous  matter  on  that  side  seemed  to  encroach  upon  the 
central  black  nucleus,  while  in  other  parts  the  penumbra  underwent  but  little 
change.  On  the  reappearance  of  the  spot  at  the  eastern  limb,  he  found  that  the 
penumbra  was  again  wanting  on  the  side  towards  the  centre  of  the  disc  ;  and  that 
when  this  part  made  its  appearance,  after  the  spot  had  advanced  a  short  distance 
upon  the  disc,  it  was  much  narrower  than  the  opposite  part.  These  various  ap- 
pearances of  the  spot  in  question  are  represented  in  Fig.  67.  Dr.  Wilson  drew 
from  these  facts  the  natural  conclusion,  that  the  spots  were  the  dark  body  of  the 

Fig.  67. 


sun  seen  through  excavations  made  in  the  luminous  matter  at  the  surface.  The 
luminous  matter  he  conceived  to  have  the  consistence  of  a  fog  or  cloud  rather  than 
of  a  liquid ;  and  suggested  that  openings  might  be  made  in  it  by  the  working  of 
some  sort  of  elastic  vapor  generated  within  the  dark  globe.  The  penumbra  sur- 
rounding each  black  spot  he  conjectured  to  be  the  sloping  sides  of  the  opening  in 
the  stratum  of  luminous  clouds.  But  according  to  this  the  penumbra  should  shade 
off  gradually  and  merge  into  the  central  black  spot  without  presenting  any  defi- 
nite line  of  demarcation  ;  whereas  its  shade  is  nearly  uniform  throughout,  and  it  is 
abruptly  terminated,  both  without  and  within.  Herschel's  theory  is  more  com- 
plete than  this,  and  differs  from  it  essentially  in  supposing  the  existence  of  an 
opake  non-lurninous  cloudy  stratum  between  the  luminous  medium  and  the  dark 
solid  globe.  It  was  devised,  after  a  long  and  diligent  inspection  of  all  the  aspects 
and  phenomena  of  the  sun's  spots,  to  account  for  these  in  all  their  varieties.  It 

S'ves  a  satisfactory  explanation  of  the  uniformity  of  shade  of  the  penumbra,  which 
r.  Wilson's  theory  does  not  do. 

409.  Herschel  conceives  the  luminous  surface  of  the  sun  to  be  constantly  in  a 
state  of  violent  agitation,  and  thai  in  comparatively  limited  districts  it  is  occasion^- 
ally  forced  up  into  masses  or  waves  of  hundreds  of  miles  in  height,  by  powerful 


PHYSICAL  CONSTITUTION  OF  THE  SUN. 


151 


upward  currents,  or  by  the  exertion  of  some  sort  of  explosive  energy  from  beneath. 
The  ridges  of  these  waves  constitute  the  faculae,  which  are  distinctly  seen  only 
when  near  the  margin  of  the  disc,  because  the  waves  there  appear  in  profile,  and 
when  near  the  middle  of  the  disc  are  seen  in  front  or  foreshortened.  This  upheav- 
ing force  is  supposed  at  times  to  acquire  such  intensity  as  to  effect  an  opening  both 
in  the  lower  and  the  upper  stratum,  and  disclose  to  view  the  dark  body  of  the  sun. 
410.  Whatever  may  be  the  true  physical  constitution  of  the  sun,  the  changes 
which  occur  upon  its  surface  take  place  with  a  rapidity  which  betokens  the  action 
of  the  most  powerful  agents,  if  not  the  existence  of  the  most  subtle  and  elastic  me- 
dia. Some  of  the  spots  are  said  to  have  closed  at  the  rate  of  nearly  a  mile  per 
second.  The  slowest  motion  noticed  is  not  far  from  a  mile  per  minute.  But  these  ve- 
locities of  approach  of  the  sides  of  a  spot  are  vastly  exceeded  by  the  rate  of  motion  o.. 
the  spots  themselves,  which  has  been  sometimes  noticed.  In  two  well-established  in- 
stances  spots  have  been  seen  to  break  into  parts,  which  have  then  rapidly  receded 
from  each  other  while  the  observer  v/as  viewing  them  through  a  telescope.  Some 
notion  of  the  stupendous  velocity  of  these  changes  may  be  obtained  from  the  con- 
sideration that  the  smallest  area  that  can  be  distinctly  discerned  upon  the  sun, 
even  through  telescopes,  is  a  circle  of  465  miles  in  diameter. 

411.  There  has  been  observed,  in  connection  with  the  sun,  at 
certain  periods  of  th.e  year,  a  faint  light  that  is  visible  before  sun- 
rise and  after  sunset,  to  which  has  been  given  the  name  of  the  Zo- 
diacal Light,  from  the  circumstance  of  its  being  mostly  compre- 
hended within  the  zodiac.  Its  color  is  white,  and  its  apparent  fig- 
ure that  of  a  spindle,  the  base  of  which  rests  on  the  sun,  and  the 
axis  of  which  lies  in  the  plane  of  the  sun's  equator  ;  such  as  would 
be  the  appearance  of  a  body  of  a  lenticular  shape,  having  its  centre 
coincident  with  the  sun  and  its  circular  edge  lying  in  the  plane  of 
the  sun's  equator.  Its  length  varies  with  the  season  of  the  year 


j?ig. 


and  the  state  of  the  atmosphere  ; 
being  sometimes  more  than  100°, 
and  at  ether  times  not  more  than 
40°  or  50°.  Its  breadth  near  the 
sun  varies  from  8°  to  30°.  It  is 
nowhere  abruptly  terminated,  but 
gradually  merges  into  the  gerferal 
light  of  the  sky.  (See  Fig.  68.) 

412.  No  generally  received  ex- 
planation of  this  singular  phenom- 
enon has  yet  been  given.  It  was 
at  one  time  supposed  to  be  the 
atmosphere  of  the  sun,  but  Laplace 
has  shown  that  this  explanation 
is  at  variance  with  the  theory  of 
gravitation.  He  found  that  at  the 
distance  of  about  sixteen  millions 
of  miles  from  the  sun's  centre 
the  centrifugal  force  balanced 
the  gravity,  and  that  therefore 
the  sun's  atmosphere  could  not  extend  beyond  this  :  but  this  dis- 
tance is  less  than  one  half  ftie  distance  of  Mercury  from  the  sun, 
whereas  the  substance  of  the  zodiacal  light  extends  beyond  the  or- 
bit of  Venus,  and  even  beyond  the  earth's  orbit. 


152  OF  THE  SUN  AM)  ITS  PHENOMENA. 

Several  theories  have  been  propounded  relative  to  the  cause  of  the  zodiacal  light 
Laplace  conceived  it  to  be  a  ring  of  nebulous,  that  is,  cloudy  and  self-luminous, 
matter,  encircling  the  sun  in  the  plane  of  his  equator.  Professor  Olmsted,  of 
New  Haven,  has  suggested  that  it  may  be  a  large  nebulous  body  revolving  around  the 
sun  in  a  regular  orbit ;  and  the  same  body  as  that  from  which  the  periodical  meteoric 
showers  are  supposed  to  proceed.  If  we  were  to  venture  another  suggestion  upon 
this  perplexing  subject,  it  would  be,  that  the  substance  of  the  zodiacal  light  may  be 
a  certain  species  of  matter  continually  in  the  act  of  flowing  away  from  the  sun 
into  free  space  :  being  expelled  by  some  repulsive  force  from  perhaps  all  parts  of 
its  surface,  but  in  much  the  greatest  quantity  from  the  region  of  the  spots,  which 
lies  about  the  equator.  Cassini,  after  an  attentive  examination  of  the  zodiacal  light 
and  the  sun's  spots  during  a  series  of  years,  conceived  that  he  had  detected  a  con- 
nection between  these  two  phenomena  ;  that  the  zodiacal  light  was  fainter  in  propor- 
tion as  the  spots  were  fewer  in  number  and  smaller.  Thus,  he  remarks,  that  after  the 
year  1688,  when  the  zodiacal  light  began  to  grow  weaker,  no  spots  appeared  upon 
the  sun.  He  thought  that  this  phenomenon  became  at  times  entirely  invisible  ;  and 
that  this  was  the  case  in  the  years  1665,  1672,  and  1681.  From  this  apparent 
connection  between  the  two  phenomena  he  drew  the  natural  conclusion,  that  the 
substance  of  the  zodiacal  light  was  some  emanation  from  the  oun's  spots.  The 
explosive  actions,  which  are  the  most  probable  cause  of  these  spots,  may  perhaps 
furnish  the  luminous  matter,  which  may  afterwards  be  driven  off  to  an  indefinite 
distance  by  some  repulsive  action  of  the  sun.  '  Certainly,  if  there  is  at  the  sun's 
surface  any  matter  of  the  same  nature'as  that  of  which  the  tails  of  comets  are  com- 
posed, it  must  be  expelled  by  the  same  repulsive  force  that  drives  off  this  species 
of  matter  from  the  heads  of  comets  and  forms  their  tails.  (See  Art.  557.) 

413.  The  zodiacal  light  is  seen  most  distinctly  in  our  northern 
climates  in  February  and  March  after  sunset,  and  in  October  and 
November  before  sunrise.  During  the  month  of  March  it  may  be 
seen  directed  towards  the  star  Aldebaran.  In  December,  though 
fainter,  it  may  often  be  seen  both  in  the  morning  and  evening. 
Also  towards  the  summer  solstice  it  is  said  to  be  discernible,  in  a 
very  pure  state  of  the  atmosphere,  both  in  the  morning  and  even- 
ing. The  reason  of  the  variations  in  the  distinctness  of  the  zodia- 
cal light,  is  found  in  the  change  of  its  inclination  to  the  horizon  at 
the  time  of  sunset  or  sunrise,  together  with  the  variation  in  the  du- 
ration of  twilight.  As  its  length  li«s  in  the  plane  of  the  sun's  equa- 
tor, its  inclination  to  the  horizon  will  be  different  like  that  of  this 
plane,  according  to  the  different  positions  of  the  sun  in  the  ecliptic. 
Since  the  sun's  equator  makes  but  a  small  angle  with  the  ecliptic, 
at  sunset,  the  zodiacal  light  will  be  most  inclined  to  the  horizon, 
and  therefore  extend  higher  up  in  the  heavens,  towards  the  vernal 
equinox,  when  the  inclination  of  the  ecliptic  to  the  hori/ou  at  sun- 
set is  at  its  maximum  ;  and,  at  sunrise,  it  will  be  most  inclined  to 
the  horizon  towards  the  autumnal  equinox,  when  the  inclination  of 
the  ecliptic  to  the  horizon  at  sunrise  is  the  greatest.  The  zodiacal 
light  is  more  easily  and  more  frequently  perceived  in  the  torrid 
zone  than  in  these  latitudes,  because  the  ecliptic  and  zodiac  make 
there  a  larger  angle  with  the  horizon,  and  because  twilight  is  of 
shorter  duration. 


PHASES  OF  THE  MOON.  153 


CHAPTER  XIV. 

OF  THE  MOON  AND  ITS  PHENOMENA. 
PHASES  OF  THE  MOON. 

414  THE  most  conspicuous  of  the  phenomena  exhibited  by  the 
moon,  is  the  periodical  change  that  is  observed  to  take  place  in  the 
form  and  size  of  its  disc.  The  different  appearances  which  the 
disc  presents  are  called  the  Phases  of  the  moon. 

The  phenomenon  in  question  is  a  simple  consequence  of  the 
revolution  of  the  moon  around  the  earth.  Let  E  (Fig.  69)  rep- 
resent the  position  of  the  earth,  ABC,  &c.,  the  orbit  of  the  moon, 

Fig.  69. 


which  we  will  suppose  for  the  present  to  lie  in  the  plane  of  the 
ecliptic,  and  ES  the  direction  of  the  sun.  As  the  distance  of  the 
sun  from  the  earth  is  about  400  times  the  distance  of  the  moon, 
lines  drawn  from  the  sun  to  the  different  parts  of  the  moon's  orbit, 
may  be  considered,  without  material  error,  as  parallel  to  each 
other.  If  we  regard  the  moon  as  an  opake  non-luminous  body, 
of  a  spherical  form,  that  hemisphere  which  is  turned  towards  the 
sun  will  continually  be  illuminated  by  him,  and  the  other  will  be 
in  the  dark.  Now,  by  virtue  of  the  moon's  motion,  the  enlightened 
hemisphere  is  presented  to  the  earth  under  every  variety  of  aspect 
in  the  course  of  a  synodic  revolution  of  the  moon.  Thus,  when 
the  moon  is  in  conjunction,  as  at  A,  this  hemisphere  is  turned 
entirely  away  from  the  earth,  and  she  is  invisible.  Soon  after 
conjunction,  a  portion  of  it  on  the  right  begins  to  be  seen,  and  as 
this  is  comprised  between  the  right  half  of  the  circle  which  limits 
the  vision,  and  the  right  half  of  the  circle  which  separates  the  en- 
lightened and  dark  hemispheres  of  the  moon,  called  the  Circle  of 
Illumination,  it  will  obviously  present  the  appearance  of  a  crescent 
with  the  horns  turned  from  the  sun,  as  represented  at  B.  As  the 
moon  advances,  more  and  more  of  the  enlightened  half  becomes 

20 


154  OF  THE  MOON  AND  ITS  PHENOMENA. 

visible,  and  thus  the  crescent  enlarges,  and  the  eastern  limb  be- 
comes less  concave.  At  the  point  C,  90°  distant  from  the  sun, 
one  half  of  it  is  seen,  and  the  disc  is  a  semi-circle,  the  eastern 
limb  being  a  right  line.  Beyond  this  point,  more  than  half  be- 
comes visible  ;  the  nearer  half  of  the  circle  of  illumination  falls  to 
the  left  of  the  moon's  centre,  as  seen  from  the  earth,  and  thus 
becomes  convex  outward.  This  phase  of  the  moon  is  repre- 
sented at  D.  When  the  moon  appears  under  this  shape,  it  is  said 
to  be  Gibbous.  In  advancing  towards  opposition,  the  disc  will 
enlarge,  and  the  eastern  limb  become  continually  more  convex ; 
and  finally  at  opposition,  where  the  whole  illuminated  face  is  seen 
from  the  earth,  it  will  become  a  full  circle.  From  opposition  to 
conjunction,  the  nearer  half  of  the  circle  of  illumination  will  form 
the  right  or  western  limb,  and  this  limb  will  pass  in  the  inverse 
order  through  the  same  variety  of  forms  as  the  eastern  limb  in 
the  interval  between  conjunction  and  opposition.  The  different 
phases  are  delineated  in  the  figure. 

415.  The  moon's  orbit  is,  ^in  fact,  somewhat  inclined  to  the 
plane  of  the  ecliptic,  instead  of  lying  in  it,  as  we  have  supposed ; 
but,  it  is  plain  that  its  inclination  cannot  change  the  order,  nor  the 
period  of  the  phases,  and  that  it  can  have  no  other  effect  than  to 
alter  somewhat  the  size  of  the  disc,  at  particular  angular  distances 
from  the  sun.     In  consequence  of  the  smallness  of  the  inclination, 
this  alteration  is  too  slight  to  be  noticed. 

416.  When  the  moon  is  in  conjunction,  it  is  said  to  be  New 
Moon;  and  when  in  opposition,  Full  Moon.     At  the  time  be- 
tween new  and  full  moon  when  the  difference  of  the  longitudes 
of  the  moon  and  sun  is  90°,  it  is  said  to  be  the  First  Quarter. 
And  at  the  corresponding  time  between  full  and  new  moon,  it  is 
said  to  be  the  Last  Quarter.     In  both  these  positions  the  moon 
appears  as  a  semi-circle,  and  is  said  to  be  dichotomized.     The 
two  positions  of  conjunction  and  opposition  are  called  Syzigies ; 
and  those  of  the  first  and  last  quarter,  Quadratures.     The  four 
points  midway  between  the  syzigies  and  quadratures  are  called 
Octants. 

417.  The  interval  from  new  moon  to  new  moon  again,  is  called 
a  Lunar  Month,  and  sometimes  a  Lunation. 

The  mean  daily  motion  of  the  sun  in  longitude  is  59'  8". 33, 
and  that  of  the  moon  13°  10'  35".03  ;  wherefore  the  moon  sepa- 
rates from  the  sun  at  the  mean  rate  of  12°  IT  26". 70  per  day; 
and  hence,  to  find  the  mean  length  of  a  lunar  month,  we  have 
the  proportion 

12°  11'  26".70  :  Id.  :  :  360°  :  x  =  29d.  12h.  44m.  2.7s. 

418.  To  determine  the  time  of  mean  new  or  full  moon  in  any 
given  month. 

Let  the  mean  longitude  of  the. sun,  and  also  the  mean  longi- 
tude of  the  moon,  at  the  beginning  of  the  year,  be  found,  and  let 


TIME  OF  NEW  OR  FULL  MOON.  155 

the  former  be  subtracted  from  the  latter,  (adding  360°  if  neces- 
sary ;)  the  remainder,  which  call  R,  will  be  the  mean  distance  of 
the  moon  to  the  east  of  the  sun,  at  the  beginning  of  the  year. 
As  the  moon  separates  from  the  sun  at  the  mean  rate  of  12°  11' 

R 

26". 70  per  day,  — ,  f  .„ — -  will  express  the  number  of  days 

1x2     11    <^o   .  /  U 

and  fractions  of  a  day,  which  at  this  epoch  have  elapsed  since  the 
last  new  moon.  This  interval  is  called  the  Astronomical  Epact. 
If  we  subtract  it  from  29d.  12h.  44m.  2.7s.  we  shall  have  the  time 
of  mean  new  moon  in  January.  This  being  known,  the  time  of 
mean  new  moon  in  any  other  month  of  the  year  results  very 
readily  from  the  known  length  of  a  lunar  month. 

The  time  of  mean  new  moon  in  any  month  being  known,  the 
time  of  mean  full  moon  in  the  same  month  is  obtained  by  the  ad- 
dition or  subtraction,  as  the  case  may  be,  of  half  a  lunar  month. 

This  problem  is  in  practice  most  easily  resolved  with  the  aid 
of  tables.  (See  Problem  XXVII.) 

419.  The  time  of  true  new  moon  differs  from  the  time  of  mean 
new  moon,  for  the  same  reasons  that  the  true  longitudes  of  the 
sun  and  moon  differ  from  the  mean.     The  same  is  true  of  the 
time  of  true  full  moon.     For  the  mode  of  computing  the  time 
of  true  new  or  full  moon  from  that  of  mean  new  or  full  moon,  see 
Problem  XXVII. 

420.  The  earth,  as  viewed  from  the  moon,  goes  through  the 
same  phases  in  the  course  of  a  lunar  month  that  the  moon  does 
to  an  inhabitant  of  the  earth.     But,  at  any  given  time,  the  phase 
of  the  earth  is  just  the  opposite  to  the  phase  of  the  moon.     About 
the  time  of  new  moon,  the  earth,  then  near  its  full,  reflects  so 
much  light  to  the  moon  as  to  render  the  obscure  part  visible. 
(See  Fig.  69.) 

MOON'S    RISING,  SETTING,  AND   PASSAGE  OVER  THE  MERIDIAN. 

421.  To  find  the  time  of  the  meridian  passage  of  the  moon  on 
a  given  day. 

Let  S  and  M  denote,  respectively,  the  right  ascension  of  the 
sun,  and  the  right  ascension  of  the  moon,  at  noon  on  the  given 
day,  and  m,  s  the  hourly  variations  of  the  right  ascension  of  the  sun 
and  moon:  also  let  t=ihe  required  time  of  the  meridian  passage. 
At  the  time  t  the  right  ascensions  will  be, 

For  the  moon     .       .       .       .     M  +  tm, 
For  the  sun        .       .       .       .      S  +  ts ; 

and,  as  the  moon  is  on  the  meridian,  the  difference  of  these  arcs 
will  be  equal  to  the  hour  angle  t ;  whence, 

*  =  M-  S+t(m  —  s)-9 
or,  if  all  the  quantities  be  expressed  in  seconds, 


156  OF  THE  MOON  AND  ITS  PHENOMENA. 

Thus,  we  find  for  the  time  of  the  meridian  passage, 

3600  (M-S) 
- 


The  quantities  M,  S,  m,  s,  arev  in  practice,  to  be  taken  from 
ephemerides  of  the  sun  and  moon. 

Example.  What  was  the  time  of  the  passage  of  the  moon's  centre  over  tin 
meridian  of  New  York,  on  the  1st  of  August,  1837  ? 

When  it  is  noon  at  New  York,  it  is  4h.  56m.  4s.  at  Greenwich.     Now,  by  the 
Nautical  Almanac, 

Aug.  1st,  at  4h.  ])  's  R.  Ascen.   .  -     .      „        8h.  58m.  36.7s. 
"          at  5h.     "        "  .        ..90      38.3 


Ih. :  56m.  4s.  :  :  2m.    1.6s.  :  1m.  53.6s. 

Aug.  1st,  at  4h.  D  's  R.  Ascen.  .        .        .        8h.  58m.  36.7s. 
Variation  of  R.  Ascen.  in  56m.  4s,      ..  1       53.6 


D  's  R.  Ascen.  at  M.  Noon  at  N.  York       .        9       0      30.3 
Aug.  1st,  0's  hourly  Variation  of  R.  Ascen.      .        .        .        9.704s 
Ih.  :  4h.  56m.  4s.  :  :  9.704s.  :  47.8s. 

Aug.  1st,  M.  Nocn  at  Greenw.,  ©'s  R.  Asc.        8h.  45m.  31.5s. 
Variation  of  R.  Ascen.  in  4h.  56m.  4s.  47.8 


©'s  R.  Ascen.  at  M.  Noon  at  N.  York        .        8    46      19.3 

Aug.  1st,  M.  Noon  at  Greenw.,  >  's  R.  Asc.       8h.  50m.  27.7s. 
Aug.  2d,      "  «:  "  9    38      J  8.7 


24)47      5J.O 

Aug.  1st,  D  's  mean  hourly  Varia.  of  R.  Asc.  1       59.6  (m) 

«         ©'s  «  «  «  9.7  0) 


m  —  s==l      49.9  =  109.9s 

By  Nautical  Almanac,  equation  of  time  =  5m.  59s. 
Ih.  :  5m.  59s.  : :  1m.,  59.6s.  :  11.9s. 

]>  's  R.  Ascen.  at  M.  Noon  at  N.  York       .        9h.    Om.  30.3s. 
Correction  for  equation  of  time  '... "  W  —  11.9 


D  's  R.  Ascen.  at  apparent  Noon  at  N.  York      9       0      18.4  (M) 
©'s  "  "  "  «  8     46      18.3  (S) 


M  —  S  =  14      0.1  =  840.1s. 
3600    .        .•'.'.        .    log.  3.55630 
M  — 8  =  840.1     ....    log.  2.92433 
3600  —  (m  —  s)  =  3490.1     .         .         ar.  co.    log.  6.45716 

Apparent  time  of  meridian  passage,  14m.  26.5s.  =  866.5s.      log.  2.93779 
Equa.  of  time  at  merid.  passage,          5      58 

Mean  time  of  meridian  passage,  Oh.  20m.  24s. 

The  Nautical  Almanac  gives  the  time  of  the  moon's  passage  over  the  meridian 
of  Greenwich  for  every  day  of  the  year.  From  this,  the  time  of  the  passage  across 
the  meridian  of  any  other  place  may  easily  be  determined,  as  follows  :  subtract  the 
time  of  the  meridian  passage  at  Greenwich  on  the  given  day,  from  that  on  the 
following  day,  and  say,  as  24h.  :  the  difference  :  :  the  longitude  of  the  place  :  a 
fourth  term.  This  fourth  term,  added  to  the  time  of  the  meridian  passage  at 


157 

Greenwich  on  the  given  day,  will  give  the  time  of  the  meridian  passage  on  the 
same  day  at  the  given  place. 

422.  Since  the  moon  has  a  motion  with  respect  to  the  sun,  the 
time  of  its  rising  and  setting  must  vary  frorn  day  to  day.     When 
first  seen  after  conjunction,  it  will  set  soon  after  the  sun.     After 
this  it  will  set  (at  a  mean)  about  50m.  later  every  succeeding 
night.     At  the  first  quarter,  it  will  set  about  midnight ;  and  at  full 
moon,  will  set  about  sunrise  and  rise  about  sunset,     louring  this 
interval  it  will  rise  in  the  daytime,  and  all  along  from  sunrise  to 
sunset.     From  full  to  new  moon,  it  will  rise  at  night  and  set 
during  the  day ;  and  the  time  of  the  rising  and  setting  will  be 
about  50m.  later  on  every  succeeding  night  and  day ;  thus,  at  the 
last  quarter  it  will  rise  about  midnight  and  set  about  midday. 

423.  The  daily  retardation  of  the  time  of  the  moon's  rising  is, 
as  just  stated,  at  a  mean,  about  50  minutes ;  but  it  varies  in  the 
course  of  a  revolution  from  about  half  an  hour  to  one  hour,  in 
these  latitudes.     The  retardation  of  the  moon's  rising  at  the  time 
of  full  moon,  varies  from  one  full  moon  to  another,  in  the  course 
of  the  year,  between  the  same  limits.     The  reason  of  these  varia- 
tions is  found  in  the  fact,  that  the  arc  of  the  ecliptic  (12°  11') 
through  which  the  moon  moves  away  from  the  sun  in  a  day,  is 
variously  inclined  to  the  horizon,  according  to  its  situation  in  the 
ecliptic,  and  therefore  employs  different  intervals  of  time  in  rising 
above  the  horizon.     This  fact  may  be  very  distinctly  shown  by 
means  of  a  celestial  globe.    It  will  be  seen  that  the  arc  in  question 
will  be  most  oblique  to  the  horizon,  and  rise  in  the  shortest  time, 
in  the  signs  Pisces  and  Aries.     Accordingly,  the  full  moons  which 
occur  in  these  signs  will  rise  with  the  smallest  retardation  fioin 
day  to  day.     These  full  moons  occur  when  the  sun  is  in  the  op- 
posite signs,  Virgo  and  Libra,  that  is,  in  September  and  October. 
They  are  called,  the  first  the  Harvest  Moon,  and  the  second  the 
Hunter's  Moon.     The  time  of  the  moon's  rising  at  these  full 
moons  will,  for  two  or  three  days,  be  only  about  half  an  hour  later 
than  on  the  preceding  day. 

424.  To  find  the  time  of  the  moon's  rising  or  setting  on  any  given  day.-  Com- 
pute the  moon's  semi-diurnal  arc  from  equation  (82),  or  (80),  according  as  it  ia 
the  time  of  the  apparent  rising  or  setting,  or  the  time  of  the  true  rising  or  setting, 
that  is  desired.     Correct  it  for  the  moon's  change  of  right  ascension  in  the  inter- 
val between  the  moon's  passage  over  the  meridian  and  setting,  by  the  following 
proportion,  24h.  :  24  -f  m  —  s  (421)  :  :  semi-diurnal  arc  :  corrected  semi-diur- 
nal arc  ;    and  add  it  to  the  time  of  the  moon's  meridian  passage,  found  as  ex- 
plained in  Art.  421.     The  result  will  be  the  time  of  the  moon's  setting ;  and 
if  this  be  subtracted  from  24  hours,  the  remainder  will  be  the  time  of  the  moon's 
rising. 

In  consequence  of  the  change  of  the  moon's  declination  in  the  interval  between 
it?  rising  and  setting,  it  would  be  more  accurate  to  compute  the  semi-diurnal  arc 
separately  for  the  moon's  rising.  In  computing  the  semi-diurnal  arc  by  equation 
(80),  the  declination  6  hours  before  or  after  the  meridian  passage  may  be  used  at 
first ;  and  afterwards,  if  a  more  accurate  result  be  desired,  the  calculation  may  be 
repeated  with  the  declination  found  for  the  computed  approximate  time.  In  equa- 
tion (81  \  R  =  refraction  —  parallax  =  33'  51"  —  57'  1"  (at  a  mean)  =  —  23'  10' 


158  OF  THE  MOON  AND  ITS  PHENOMENA. 


ROTATION  AND  LIBRATIONS  OF  THE  MOON. 

425.  The  moon  presents  continually  nearly  the  same  face  to- 
wards the  earth ;  for,  the  same  spots  are  always  seen  in  nearly 
the  same  position  upon  the  disc.     It  follows,  therefore,  that  it 
rotates  on  its  axis  in  the  same  direction,  and  with  the  same  angu- 
lar velocity,  or  nearly  so,  that  it  revolves  in  its  orbit,  and  thus 
completes  one  rotation  in  the  same  period  of  time  in  which  it  ac- 
complishes a  revolution  in  its  orbit. 

426.  The  spots  on  the  moon's  disc,  although  they  constantly 
preserve  very  nearly  the  same  situations,  are  not,  however,  strictly 
stationary.     When  carefully  observed,  they  are  seen  alternately 
to  approach  and  recede  from  the  edge.     Those  that  are  very  near 
the  edge  successively  disappear  and  again  become  visible.     This 
vibratory  motion  of  the  moon's  spots  is  called  Libration. 

427.  There  are  three  librations  of  the  moon,  that  is,  a  vibratory 
motion  of  its  spots  from  three  distinct  causes. 

(1.)  The  moon's  motion  of  rotation  being  uniform,  small  portions 
on  its  east  and  west  sides  alternately  come  into  sight  and  disap- 
pear, in  consequence  of  its  unequal  motion  in  its  orbit.  The 
periodical  oscillation  of  the  spots  in  an  easterly  and  westerly  direc- 
tion from  this  cause,  is  called  the  Libration  in  Longitude. 

(2.)  The  lunar  spots  have  also  a  small  alternate  motion  from 
north  to  south.  This  is  called  the  Libration  in  Latitude,  and  > 
accounted  for  by  supposing  that  the  moon's  axis  is  not  exactly 
perpendicular  to  the  plane  of  its  orbit,  and  that  it  remains  contin- 
ually parallel  to  itself.  On  this  supposition  we  ought  sometimes 
to  see  beyond  the  north  pole  of  the  moon,  and  sometimes  beyond 
the  south  pole. 

(3.)  Parallax  is  the  cause  of  a  third  libration  of  the  moon.  The 
spectator  upon  the  earth's  surface  being  removed  from  its  centra, 
the  point  towards  which  the  moon  continually  presents  the  same 
hemisphere,  he  will  see  portions  of  the  moon  a  little  different 
according  to  its  different  positions  above  the  horizon.  The  diur- 
nal motion  of  the  spots  resulting  from  the  parallax,  is  called  the 
Diurr>sd  or  Parallactic  Libration. 

428.  The  exact  position  of  the  moon's  equator,  like  that  of  the 
sun's,  is  derived  from  accurate  observations  of  the  situations  of 
the  spots  upon  the  disc.     From  calculations  founded  upon  such 
observations,  it  has  been  ascertained  that  the  plane  of  the  moon's 
equator  is  constantly  inclined  to  the  plane  of  the  ecliptic  under  an 
angle  of  1°  30',  and  Intersects  it  in  a  line  which  is  always  parallel 
to  the  line  of  the  nodes.     It  follows  from  the  last-mentioned  cir- 
cumstance, that  if  a  plane  be  supposed  to  pass  through  the  centre 
of  the  moon,  parallel  to  the  ecliptic,  it  will  intersect  the  f>lane  of 
the  moon's  equator  ajiti  that  of  its  orbit  in  the  same  line  in  which 
these  planes  intersect  each  other.     The  plane  in  question  will  lie 
between  the  plane,  of  the  equator  and  that  of  the  orbit.     Tt  will 


159 

make  with  the  first  an  angle  of  1°  30',  and  with  the  second  an 
angle  cf  5°  9'. 

DIMENSIONS  AND  PHYSICAL  CONSTITUTION  OF  THE  MOON. 

429.  The  phases  of  the  moon  prove  it  to  be  an  opake  spherical 
body.     Its  diameter  is  found  by  means  of  equation  (83),  viz  : 


where  d  denotes  the  diameter  sought,  R  the  radius  of  the  earth,  6 
the  apparent  diameter  of  the  moon  at  a  given  distance,  and  H  its 
horizontal  parallax  at  the  same  distance. 

The  greatest  equatorial  horizontal  parallax  of  the  moon  is  61' 
24",  and  the  corresponding  apparent  diameter  33'  31"  :  thus  we 
have 

33'  31"  3 

=  2R        ,  48//  =  2R  ^  (very  nearly)  -  2161  miles. 


The  diameter  of  the  moon  being  to  the  diameter  of  the  earth 
as  3  to  11,  the  surface  of  the  moon  is  to  the  surface  of  the  earth 
as  32  to  II2,  or  as  1  to  13  ;  and  the  volume  of  the  moon  is  to  the 
volume  of  the  earth  as  33  to  II3,  or  as  1  to  49. 

430.  When  the  moon  is  viewed  with  a  telescope,  the  edge  of 
the  disc,  which  borders  upon  the  dark  portion  of  the  face,  is  seen 
to  be  very  irregular  and  serrated,  (see  Fig.  70.)  It  is  "hence  in 

Fig.  70. 


ferred  that  the  surface  of  the  moon  is  diversified  with  mountains 
and  valleys.  The  truth  of  this  inference  is  confirmed  by  the  fact 
that  bright  insulated  spots  are  frequently  seen  on  the  dark  part  of 
the  face  near  the  edge  of  the  disc,  which  gradually  enlarge  until 
they  become  united  to  it.  These  bright  spots  are  doubtless  the 
tops  of  mountains  illuminated  by  the  sun,  while  the  surrounding 


160  OF  THE  MOON  AND  ITS  PHENOMENA. 

regions  that  are  less  elevated  are  involved  in  darkness.  The  disc 
is  also  diversified  with  spots  of  different  shapes  and  different  de- 
grees of  brightness.  The  brighter  parts  are  supposed  to  be  ele- 
vated land,  and  the  dark  to  be  plains,  and  valleys,  or  cavities. 

431 .  The  number  of  the  lunar  mountains  is  very  great.     Many 
of  them,  by  their  form  and  grouping,  furnish  decided  indications 
of  a  volcanic  origin. 

From  measurements  made  with  the  micrometer,  of  the  lengths 
of  their  shadows,  or  of  the  distance  of  their  summits  when  first 
illuminated,  from  the  adjacent  boundary  of  the  disc,  the  heights  of 
a  number  of  the  lunar  mountains  have  been  computed.  Accord- 
ing to  Herschel,  the  altitude  of  the  highest  is  only  about  1|  Eng- 
lish miles.  But  Schroeter  of  Lilienthal,  a  distinguished  Seleno- 
graphist,  makes  the  elevation  of  some  of  the  lunar  mountains  to 
exceed  5  miles:  and  the  more  recent  measurements  of  MM.  Baer 
and  Madler  of  Berlin  lead  to  similar  results. 

432.  There  are  no  seas  nor  other  bodies  of  water  upon  the  sur- 
face of  the  moon.     Certain  dark  and  apparently  level  parts  of  the 
moon  were  for  some  time  supposed  to  be  extended  sheets  of  wa- 
ter, and,  under  this  idea,  were  named  by  Hevelius  Mare  Imbrium, 
Mare  Crisium,  &c. :  but  it  appears  that  when  the  boundary  of 
light  and  darkness  falls  upon  these  supposed  seas,  it  is  still  more 
or  less  indented  at  some  points,  and  salient  at  others,  instead  of 
being,  as  it  should  be,  one  continuous  regular  curve ;  besides, 
when  these  dark  spots  are  viewed  with  good  telescopes,  they  are 
found  to  contain  a  number  of  cavities,  whose  shadows  are  dis- 
tinctly perceived  falling  within  them.     The  spots  in  question  are 
therefore  to  be  regarded  as  extensive  plains  diversified  by  mode- 
rate elevations  and  depressions.     The  entire  absence  of  water  also 
from  the  farther  hemisphere  of  the  moon  may  be  inferred  from  the 
fact  that  the  moon's  face  is  never  obscured  by  clouds  or  mists. 

433.  It  has  long  been  a  question  among  Astronomers,  whether  the  moon  has  an 
atmosphere.  It  ?s  asserted,  that,  if  it  has  any,  it  must  be  exceedingly  rare,  or 
very  limited  in  its  extent,  since  it  does  not  sensibly  diminish  or  refract  the  light 
of  a  star  seen  in  contact  with  the  moon's  limb  ;  for  when  a  star  experiences  an 
occultalion  by  reason  of  the  interposition  of  the  moon  between  it  and  the  eye  of 
the  observer,  it  does  not  disappear  or  undergo  any  diminution  of  lustre  until  the 
body  of  the  moon  reaches  it,  and  the  duration  of  the  occupation  is  as  it  is  com- 
puted, without  making  any  allowance  for  the  refraction  of  a  lunar  atmosphere. 
But  it  is  maintained,  on  the  other  hand,  that  these  facts,  if  allowed,  are  not  op- 
posed  to  the  supposition  of  the  existence  of  an  atmosphere  of  a  few  miles  only  in 
height ;  and  that  certain  phenomena  which  have  been  observed  afford  indubitable 
evidence  of  the  presence  of  a  certain  limited  body  of  air  upon  the  moon's  surface. 
Thus  the  celebrated  Schroeter,  in  the  course  of  some  delicate  observations  made 
upon  the  crescent  moon,  perceived  a  faint  grayish  light  extending  from  the  horns 
of  the  crescent  a  certain  distance  into  the  dark  part  of  the  moon's  face.  This  he 
conceived  to  be  the  moon's  twilight,  and  hence  inferred  the  existence  of  a  lunar 
atmosphere.  From  the  measurements  which  he  made  of  the  extent  of  this  light 
he  calculated  the  height  of  that  portion  of  the  atmosphere  which  was  capable  of 
affecting  the  light  of  a  star  to  be  about  one  mile.  Again,  in  total  eclipses  of  the 
gun,  occasioned  by  the  interposition  of  the  moon,  the  dark  body  of  the  moon  has 
been  ceen  surrounded  by  a  luminous  ring,  which  was  at  first  »nost  distinct  at  the 


161 

part  where  the  sun  was  last  seen,  and  afterwards  at  the  part  where  the  first  ray 
darted  from  the  sun.  This  is  supposed  to  have  been  a  lunar  twilight.  A  similar 
phenomenon  was  observed  in  the  annular  eclipse  of  1836,  just  before  the  comple« 
tion  of  the  ring,  at  the  po.nt  where  the  junction  took  place. 

On  the  whole,  it  seems  most  probable  that  the  moon  has  a  smaU  atmosphere. 


434.  The  surface  of  the  moon,  like  that  of  the  earth,  presents  the  two  general 
varieties  of  level  and  mountainous  districts  ;  but  it  differs  from  the  earth's  surface 
in  having  no  seas,  or  other  bodies  of  water,  upon  it,  (432,)  and  in  being  more  rug- 
ged and  mountainous.  The  comparatively  level  regions  occupy  somewhat  more 
than  one-third  of  the  nearer  half  of  the  moon's  surface.  These  are,  in  general, 
the  darker  parts  of  the  disc.  The  lunar  plains  vary  in  extent  from  40  or  50  miles 
to  700  miles  in  diameter.  The  mountainous  formations  of  the  other  parts  of  the 
surface  offer  three  marked  varieties,  viz : 

(1.)  Insulated  Mountains,  which  rise  from  plains  nearly  level,  and  which  may 
be  supposed  to  present  an  appearance  somewhat  similar  to  Mount  Etna  or  the 
Peak  of  Teneriffe.  The  shadows  of  these  mountains,  in  certain  phases  of  the 
moon,  are  as  distinctly  perceived  as  the  shadow  of  an  upright  staff  when  placed 
opposite  to  the  sun.*  The  perpendicular  altitudes  of  some  of  them,  as  deter- 
mined from  the  lengths  of  their  shadows,  are  between  four  and  five  miles.  Insu- 
lated mountains  frequently  occur  in  the  centres  of  circular  plains.  They  are 
then  called  Central  Mountains. 

(2.)  Ranges  of  Mountains,  extending  in-  length  two  or  three  hundred  miles. 
These  ranges  bear  a  distinct  resemblance  to  our  Alps,  Appenines,  and  Andes,  but 
they  are  much  less  in  extent,  and  do  not  form  a  very  prominent  feature  of  the 
lunar  surface.  Some  of  them  appear  very  rugged  and  precipitous,  and  the  highest 
ranges  are,  in  some  places,  above  four  miles  in  perpendicular  altitude.  In  some 
instances  thev  run  nearly  in  a  straight  line  from  northeast  to  southwest,  as  in  that 
range  called  the  Appenines ;  in  other  cases  they  assume  the  form  of  a  semicircle 
or  a  crescentt 

(3.)  Circular  Formations.  The  general  prevalence  of  this  remarkable  class  of 
mountainous  formations  is  the  great  characteristic  feature  of  the  topography  of 
the  moon's  surface.  It  is  subdivided  by  late  selenographists  into  three  orders,  viz : 
Walled  Plains,  whose  diameter  varies  from  one  hundred  and  twenty  to  forty  or 
fifty  miles  ;  Ring  Mountains,  the  diameter  of  which  descends  to  ten  miles ;  and 
Craters,  which  are  still  smaller.  The  term  crater  is  sometimes  extended  to  all 
the  varieties  of  circular  formations.  They  are  also  sometimes  called  Caverns,  be- 
cause their  enclosed  plains  or  bottoms  are  sunk  considerably  below  the  general 
level  of  the  moon's  surface. 

The  different  orders  of  the  circular  formations  differ  essentially  from  each  other 
only  in  size.  The  principal  features  of  their  constitution  are,  for  the  most  part, 
the  same,  and  they  present  similar  varieties.  Sometimes  terraces  are  seen  going 
round  the  ^hole  ring.  At  other  times  ranges  of  concentric  mountains  encircle 
the  inner  toot  of  the  wall,  leaving  intermediate  valleys.  Again,  we  have  a  few 
ridges  of  low  mountains  stretching  through  the  circle  contained  by  the  wall,  but 
oftener  isolated  conical  peaks  start  up,  and  very  frequently  small  craters  having 
on  an  inferior  scale  every  attribute  of  the  large  one-t  The  smaller  craters, 
however,  offer  some  characteristic  peculiarities.  Most  of  them  are  without  a 
flat  bottom,  and  have  the  appearance  of  a  hollow  inverted  cone  with  the  sides 
tapering  towards  the  centre.  Some  have  no  perceptible  outer  edge,  their  margin 
being  on  a  level  with  the  surrounding  regions  :  these  are  called  Pits. 

Tho  bounding  ridge  of  the  lunar  craters  or  caverns  is  much  more  precipitous 
within  than  without ;  and  the  internal  depth  of  the  crater  is  always  much  lower 
than  the  general  surface  of  the  moon.  The  depth  varies  from  one-third  of  a  mile 
to  three  miles  and  a  half. 

These  curious  circular  formations  occur  at  almost  every  part  of  the  surface,  but 
are  most  abundant  in  the  southwestern  regions.  It  is  the  strong  reflection  of  their 

*  Dick's  Celestial  Scenery,  p.  256.  t  Ibid.  p.  257. 

t  Nichol's  Phenomena  of  the  S  ;lar  System,  p.  167. 
21 


162  ECLIPSES  OF  THE  SUN  AND  MOON. 

mountainous  ridges  which  gives  to  that  part  of  the  moon's  surface  its  superior  lus- 
tre.    The  smaller  craters  occupy  nearly  two-fifths  of  the  moon's  visible  surface. 


CHAPTER   XV. 

ECLIPSES  OF  THE  SUN  AND  MOON. OCCULTATIONS  OF  THE 

FIX.ED  STARS. 

435.  AN  eclipse  of  a  heavenly  body  is  a  privation  of  its  light 
occasioned  by  the  interposition  of  some  opake  body  between  it 
and  the  eye,  or  between  it  and  the  sun.     Eclipses  are  divided, 
with  respect  to  the  objects  eclipsed,  into  eclipses  of  the  sun, 
Fig.  71.  of  the  moon,  and   of  the   satellites, 

(334  ;)  and,  with  respect  to  circum- 
stances,  into  total,  partial,  annular, 
d  central.  A  total  eclipse  is  one  in 
which  the  whole  disc  of  the  lumi- 
nary is  darkened ;  a  partial  one  is 
when  only  a  part  of  the  disc  is  dark- 
ened. In  an  annular  eclipse  the 
whole  is  darkened,  except  a  ring  or 
annulus,  which  appears  round  the 
dark  part  like  an  illuminated  border; 
the  definition  of  a  central  eclipse  will 
be  given  in  another  place. 

ECLIPSES  OF  THE  MOON. 

436.  An  eclipse  of  the  moon  is  oc- 
casioned by  an  interposition  of  the 
body  of  the  earth  directly  between  the 
sun  and  moon,  and  thus  intercepting 
the  light  of  the  sun  ;  or  the  moon  is 
eclipsed  when  it  passes  through  part 
of  the  shadow  of  the  earth,  as  pro- 
jected from  the  sun.     Hence  it  is  ob- 
vious that  lunar  eclipses  can  happen 
only  at  the  time  of  full  moon,  for  it 
is  then  only  that  the  earth  can  be  be- 
tween the  moon  and  the  sun. 

437.  Since  the  sun  is  much  larger 
than  the  earth,  the  shadow  of  the  earth 
must  have  the  form  of  a  cone,  the  length 
of  which  will  depend  on  the  relative 
magnitudes  of  the  two  bodies  and  their 

distance  fn>m~~each  other.      Let  the  circles  AGB,  agb,  (Fig.  71,) 


163 

be  sections  of  the  sun  and  earth  by  a  plane  passing  through  their 
centres  S  and  E  ;  Aa,  B6,  tangents  to  these  circles  on  the  same 
side,  and  Ac?,  Be,  tangents  on  different  sides.  The  triangular  space 
uCb  will  be  a  section  of  the  earth's  shadow  or  Umbra,  as  it  is 
sometimes  called.  The  line  EC  is  called  the  Axis  of  the  Shadow. 
If  we  suppose  the  line  cp  to  revolve  about  EC,  and  form  the  sur- 
face of  the  frustrum  of  a  cone,  of  which  pcdq  is  a  section,  the 
space  included  within  that  surface  and  exterior  to  the  umbra,  is 
called  the  Penumbra.  It  is  plain  that  points  situated  within  the 
umbra  will  receive  no  light  from  the  sun  ;  and  that  points  situated 
within  the  penumbra  will  receive  light  from  a  portion  of  the  sun's 
disc,  and  from  a  greater  portion  the  more  distant  they  are  from  the 
umbra. 

438.  To  find  the  length  of  the  earth's  shadow.  —  Let  L=  the 
length  of  the  shadow  ;  R=the  radius  of  the  earth  ;  <5  =  sun's  ap- 
parent semi-diameter,  and  p  =  sun's  parallax.  The  right-angled 
triangle  EaC  (Fig.  71)  gives 


sin  ECa* 
andECa  =  SEA  —  EAC  =  5  —  p;  whence, 

.  .  .  (86.) 


—  p) 

p)  is  only  a 
its  sine,  and  there 


As  the  angle  (<5  —  p)  is  only  about  16',  it  will  differ  but  little  from 
fore, 


(nearly); 


or,  if  5  and  p  be  expressed  in  seconds, 


,       ^,. 

L==R  —  r  --  (nearly)  .  .  .  (87). 

The  shadow  will  obviously  be  the  shortest  when  the  sun  is  the 
nearest  to  the  earth.  We  then  have  8  =  W  18",  andp  =  9",  which 
gives  L  =  213R.  The  greatest  distance  of  the  moon  is  a  little 
less  than  64  R,  It  appears,  then,  that  the  earth's  sJiadow  always 
extends  to  more  than  three  times  the  distance  of  the  moon. 

439.  Let  kMh  be  a  circular  arc,  described  about  E  the  centre 
of  the  earth,  and  with  a  radius  equal  to  the  distance  between  the 
centres  of  the  earth  and  moon  at  the  time  of  opposition.     The  an- 
gle MEw,  the  apparent  semi-diameter  of  a  section  of  the  earth's 
shadow,  made  at  the  distance  of  the  moon's  centre,  is  called  the 
Semi-diameter  of  the  Earth's  Shadow.    And  the  angle  ME  A,  the 
apparent  semi-diameter  of  a  section  of  the  penumbra,  at  the  same 
distance,  is  called  the  Semi-diameter  of  the  Penumbra. 

440.  Were  the  plane  of  the  moon's  orbit  coincide™  with  the 
plane  of  the  ecliptic,  there  would  be  a  lunar  eclipse  at  every  full 
moon  ;  out,  as  it  is  inclined  tc  't,  an  eclipse  can  happen  only  when 


164 


ECLIPSES  OF  THE  SUN  AND  MOON. 


Fig.  72. 


-TNf 


the  full  moon  takes  place  either  in  one 
of  the  nodes  of  the  moon's  orbit,  or  so 
near  it  that  the  moon's  latitude  does  not 
exceed  the  sum  of  the  apparent  semi-di- 
ameters of  the  moon  and  of  the  earth's 
shadow.  This  will  be  better  understood 
on  referring  to  Fig.  72,  in  which  N'C 
represents  a  portion  of  the  ecliptic,  and 
N'M  a  portion  of  the  moon's  orbit,  N' 
the  descending  node,  E  the  earth,  ES, 
ES',  ES"  three  different  directions  of 
the  sun,  s,  s',  s"  sections  of  the  earth's 
shadow  in  the  three  several  positions 
0\\  corresponding  to  these  directions  of  the 
\  »  N  v  sun,  and  m,  m',  m"  the  moon  in  opposi- 
\tion.  It  will  be  seen  that  the  moon 
\,  \-  will  not  pass  into  the  earth's  shadow 
1  unless  at  the  time  of  opposition  it  is 
nearer  to  the  node  than  the  point  m',  where  the  latitude  rn's'  is 
equal  to  the  sum  of  the  semi-diameters  of  the  moon  and  shadow. 

441.  To  determine  the  distance  from  the  node,  beyond  which 
there  can  be  no  eclipse,  we  must  ascertain  the  semi-diameter  of  the 
earth's  shadow.     Let  this  be  denoted  by  A,  and  let  P  =  the  moon's 
parallax. 

MEm  =  Ema  -  ECm  (Fig.  71) ; 

but  Ema  =  P  and  ECm  =  6  —p  (438) ;  therefore, 
MEw=A  =  P+p-<5  .  .  .  (88). 

The  semi-diarneter  of  the  shadow  is  the  least  when  the  moon  is 
in  its  apogee  and  the  sun  is  in  its  perigee,  or  when  P  has  its  mini- 
mum, and  <5  its  maximum  value.  In  these  positions  of  the  moon 
and  sun,  P  =  53'  48",  d=W  18",  and  p  =  9".  Substituting,  we 
obtain  for  the  least  semi-diameter  of  the  earth's  shadow  37'  39". 
and  for  its  least  diameter  1°  15'  18".  The  greatest  apparent  diam- 
eter of  the  moon  is  33'  31".  Whence  it  appears,  that  the  diameter 
of  the  earth's  shadow  is  always  more  than  twice  the  diameter  of 
the  moon. 

The  mean  values  of  P  and  <5  are  respectively  57'  1",  and  16'  1"; 
which  gives  for  the  mean  semi-diameter  of  the  earth's  shadow 
41'  9". 

442.  If  to  P  +j9  —  <5,  the  semi-diameter  of  the  earth's  shadow, 
we  add  d,  the  semi-diameter  of  the  moon,  the  sum  P  +  p  +  d  —  <5 
will  express  the  greatest  latitude  of  the  moon  in  opposition,  at  which 
an  eclipse  can  happen. 

It  is  easy  for  a  given  value  of  P  -\-p  +  d  —  8,  and  for  a  given  in- 
clination of  the  moon's  orbit,  to  determine  within  what  distance  from 
the  node  the  moon  must  be  in  order  that  an  eclipse  may  take  place. 
By  taking  the  least  and  greatest  inclinations  of  the  orbit,  the  great- 


LUNAR  ECLIPTIC  LIMITS.  165 

est  and  least  values  of  P  -f  p  +  d—  <$,  and  also  taking  into  view  the 
inequalities  in  the  motions  of  the  sun  and  moon,  it  has  been  found, 
that  when  at  the  time  of  mean  full  moon  the  difference  of  the  mean 
longitudes  of  the  moon  and  node  exceeds  13°  21',  there  cannot  be 
an  eclipse  ;  but  when  this  difference  is  less  than  7°  47'  there  must 
be  one.  Between  7°  47'  and  13°  21'  the  happening  of  the  eclipse 
is  doubtful.  These  numbers  are  called  the  Lunar  Ecliptic  Limits. 

To  determine  at  what  full  moons  in  the  course  of  any  one  year 
there  will  be  an  eclipse,  find  the  time  of  each  mean  full  moon, 
(418) ;  and  for  each  of  the  times  obtained  find  the  mean  longitude 
of  the  sun,  and  also  of  the  moon's  node,  and  compare  the  differ- 
ence of  these  with  the  lunar  ecliptic  limits.  Should,  however,  the 
difference  in  any  instance  fall  between  the  two  limits,  farther  cal- 
culation will  be  necessary. 

This  problem  may  be  solved  more  expeditiously  by  means  of 
tables  of  the  sun's  mean  motion  with  respect  to  the  moon's  node. 
(See  Prob.  XXVIII.) 

443.  The  magnitude  and  duration  of  an  eclipse  depend  upon  the 
proximity  of  the  moon  to  the  node  at  the  time  of  opposition.     In 
order  that  the  centre  of  the  moon  may  be  on  the  same  right  line 
with  the  centres  of  the  sun  and  earth,  or,  in  technical  language, 
that  a  central  eclipse  may  happen,  the  opposition  must  take  place 
precisely  in  the  node.    A  strictly  central  eclipse,  therefore,  seldom, 
if  ever,  occurs.    As  the  mean  semi-diameter  of  the  earth's  shadow 
is  41'  9"  (441),  the  mean  semi-diameter  of  the  moon  15'  33",  and 
the  mean  hourly  motion  of  the  moon  with  respect  to  the  sun  30' 
29",  the  mean  duration  of  a  central  eclipse  would  be  about  3fh. 

444.  Since  the  moon  moves  from  west  to  east,  an  eclipse  of  the 
moon  must  commence  on  the  eastern  limb,  and  end  on  the  western. 

445.  In  the  investigations  in  Arts.  438,  441,  we  have  supposed 
the  cone  of  the  earth's  shadow  to  be  formed  b}^  lines  drawn  from 
the  edge  of  the  sun,  and  touching  the  earth's  surface.    This,  prob- 
ably, is  not  the  exact  case  of  nature  ;  for  the  duration  of  the  eclipse, 
and  thus  the  apparent  diameter  of  the  earth's  shadow,  is  found  by 
observation  to  be  somewhat  greater  than  would  result  from  this 
supposition.     This  circumstance  is  accounted  for  by  supposing 
those  solar  rays  that,  from  their  direction,  would  glance  by  and  rase 
the  earth's  surface,  to  be  stopped  and  absorbed  by  the  lower  strata 
of  the  atmosphere.     In  such  a 'case  the  conical  boundary  of  the 
earth's  shadow  would  be  formed  by  certain  rays  exterior  to  the 
former,  and  would  be  larger. 

The  moon  in  approaching  and  receding  from  the  earth's  total 
shadow,  or  umbra,  passes  through  the  penumbra,  and  thus  its  light, 
instead  of  being  extinguished  and  recovered  suddenly,  experiences 
at  the  beginning  of  the  eclipse  a  gradual  diminution,  and  at  the  end 
a  gradual  increase.  On  this  account  the  times  of  the  beginning 
and  end  of  the  eclipse  cannot  be  noted  with  precision,  and  in  con- 
sequence astronomers  differ  as  to  the  amount  of  the  increase  in  the 


166 


ECLIPSES  OF  THE  SUN  AND  MOON. 


size  of  the  earth's  shadow  from  the  cause  above  mentioned.  It  is 
the  practice,  however,  in  computing  an  eclipse  of  the  moon,  to  in- 
crease the  semi-diameter  of  the  shadow  by  a  ^\  part ;  or,  which 
amounts  to  the  same,  to  add  as  many  seconds  as  the  semi-diamete~ 
contains  minutes. 

446.  It  is  remarked  in  total  eclipses  of  the  moon,  that  the  mooa 
is  not  wholly  invisible,  but  appears  with  a  dull  reddish  light. 

This  phenomenon  is  doubtless  another  effect  of  the  earth's  at- 
mosphere, though  of  a  totally  different  nature  from  the  preceding. 
Certain  of  the  sun's  rays,  instead  of  being  stopped  and  absorbed, 
are  bent  from  their  rectilinear  course  by  the  refracting  power  of 
the  atmosphere,  so  as  to  form  a  cone  of  faint  light,  interior  to  that 
cone  which  has  been  mathematically  described  as  the  earth's  shad- 
ow, which  falling  upon  the  moon  renders  it  visible. 

447.  As  an  eclipse  of  the  moon  is  occasioned  by  a  real  loss  of 
its  light,  it  must  begin  and  end  at  the  same  instant,  and  present 
precisely  the  same  appearance,  to  every  spectator  who  sees  the 
moon  above  his  horizon  during  the  eclipse.     If  will  be  shown  that 
the  case  is  different  with  eclipses  of  the  sun. 

CALCULATION  OF  AN  ECLIPSE  OF  THE  MOON. 

448.  The  apparent  distance  of  the  centre  of  the  moon  from  the 
axis  of  the  earth's  shadow,  and  the  arcs  passed  over  by  the  centre 
of  the  moon  and  the  axis  of  the  shadow  during  an  eclipse  of  the 
moon,  being  necessarily  small,  they  may,  without  material  error, 
be  considered  as  right  lines.     We  may  also  consider  the  apparent 
motion  of  the  sun  in  longitude,  and  the  motions  of  the  rnoon  in 
longitude  and  latitude,  as  uniform  during  the  eclipse.    These  sup- 
positions being  made,  the  calculation  of  the  circumstances  of  an 
eclipse  of  the  moon  is  very  simple. 

Fig.  73 


Let  NF  (Fig.  73)  be  a  part  of  the  ecliptic,  N  the  moon's  as 
cending  node,  NL  a  part  of  the  moon's  orbit,  C  the  centre  of  a 
section  of  the  earth's  shadow  at  the  moon,  CK  perpendicular  to 
NF  a  circle  of  latitude,  and  C'  the  centre  of  the  moon  at  the  in 
stant  of  opposition :  then  CC',  which  is  the  latitude  of  the  moon 
in  opposition,  is  the  distance  of  the  centres  of  the  shadow  and 
moon  at  that  time.     The  moon  and  shadow  both  have  a  motion, 
and  in  the  same  direction,  as  from  N  towards  F  and  L.     It  is  the 


167 

practice,  however,  to  regard  the  shadow  as  stationary,  and  to  attri- 
bute to  the  moon  a  motion  equal  to  the  relative  motion  of  the  moon 
and  shadow.  Tfce  orbit  that  would  be  described  by  the  moon's 
centre  if  it  had  such  a  motion,  is  called  the  Relative  Orbit  of  the 
moon.  Inasmuch  as  the  circumstances  of  the  eclipse  depend  al- 
together upon  the  relative  motion  of  the  moon  and  shadow,  this 
mode  of  proceeding  is  obviously  allowable. 

As  the  shadow  has  no  motion  in  latitude,  the  relative  motion  of 
the  moon  and  shadow  in  latitude  will,  be  equal  to  the  moon's  ac- 
tual motion  in  latitude  :  and  since  the  centre  of  the  earth's  shadow 
moves  in  the  plane  of  the  ecliptic  at  the  same  rate  as  the  sun,  the 
relative  motion  of  the  moon  and  shadow  in  longitude  will  be  equal 
to  the  difference  between  the  motions  of  the  sun  and  moon  in  lon- 
gitude. We  obtain,  therefore,  the  relative  position  of  the  centres 
of  the  moon  and  shadow  at  any  interval  t,  following  opposition,  by 
laying  off  Cm  equal  to  the  difference  of  the  motions  of  the  sun  and 
moon  in  longitude  in  this  interval,  through  m  drawing  rnM.  per- 
pendicular to  NF,  and  cutting  off  mM.  equal  to  the  latitude  at  op- 
position plus  the  motion  in  latitude  in  the  interval  t  :  M  will  be  the 
position  of  the  moon's  centre  in  the  relative  orbit,  the  centre  of  the 
shadow  being  supposed  to  be  stationary  at  C.  As  the  motion  of 
the  sun  in  longitude,  and  of  the  moon  in  longitude  and  latitude,  is 
considered  uniform,  the  ratio  of  C'm'  (=  Cm,  the  difference  be^ 
tween  the  motions  of  the  sun  and  moon  in  longitude)  to  Mm'  the 
moon's  motion  in  latitude,  is  the  same,  whatever  may  be  the  length 
of  the  interval  considered.  It  follows,  therefore,  that  the  relative 
orbit  of  the  moon  N'C'M  is  a  right  line. 

449.  The  relative  orbit  passes  through  C',  the  place  of  the  moon's  centre  at  op- 
position  :  its  position  will  therefore  be  known,  if  its  inclination  to  the  ecliptic  be 
found.     Now  we  have 

Mm'  moon's  motion  in  latitude 

tan  mc.lma.  ~  —  —  ;  =  -  :  ----  - 
Cm        moon's  mot.  in  long.  —  sun's  mot.  in  long. 

450.  The  following  data  arc  requisite  in  the  calculation  of  the  circumstances  of 
a  lunar  eclipse  : 

T  =  time  of  opposition. 

M  =  moon's  hourly  motion  in  longitude. 

n  =  moon's  hourly  motion  in  latitude. 

m  =  sun's  hourly  motion  in  longitude. 

X  =  moon's  latitude  at  opposition. 

d  =  moon's  semi-diameter. 

&  =  sun's  semi-diameter. 

P  =  moon's  horizontal  parallax 

p  =  sun's  horizontal  parallax. 

s  =  semi-diameter  of  earth's  shadow. 

I  =  inclination  of  relative  orbit. 

h  =  moon's  hourly  motion  on  relative  orbit. 

T,  M,  n,  m,  X,  d,  S,  P,  and  p,  are  derived  from  Tables  of  the  sun  and  moon. 
(See  Problems  IX  and  XIV.) 

The  quantities  s,  I,  and  h,  may  be  determined  from  these  : 

—  6)  (441  and  445)  .  .  .  (89)} 


tang  I  =  (449)  .  .  .  (90). 


168 

The  triangle  C'Mm'  gives 


ECLIPSES    OP   THE    SUN   AND   MOON. 


C'M 


cos  MOW 


M  — m 


(91). 


451.  The  above  quantities  being  supposed  to  be  known,  let  N'CF  (Fig.  74)  re- 
present the  ecliptic,  and  C   the  stationary  centre  of  the  earth's  shadow.     Let 


Fig.  74. 


_K  S 


CC'  =  X,  and  let  N'C'L'  represent  the  relative  orbit  of  the  moon.  We  here  sup- 
pose the  moon  to  be  north  of  the  ecliptic  at  the  time  of  opposition,  and  near  its 
ascending  node :  when  it  is  south  of  the  ecliptic  X  is  to  be  laid  off  below  N'CF, 
and  when  it  is  approaching  either  node,  the  relative  orbit  is  inclined  to  the  right. 
Let  the  circle  KFK'R,  described  about  the  centre  C,  represent  the  section  of  the 
earth's  shadow  at  the  moon ;  and  let  /,  /',  and  g,  g1,  be  the  respective  places  of  the 
moon's  centre,  at  the  beginning  and  end  of  the  eclipse,  and  at  the  beginning  and 
end  of  the  total  eclipse.  C/  =  C/ '  =  s  -f  d,  and  Cg  =  Cg'  =  s  —  d.  Draw  CM 
perpendicular  to  N'C'L',  and  M  will  represent  the  place  of  the  moon's  centre  when 
nearest  the  centre  of  the  shadow :  it  will  also  be  its  place  at  the  middle  of  the 
eclipse  ;  for  since  C/  =  C/,  and  CM  is  perpendicular  to  N'C'/',  M/  =  M/. 

452.  Middle  of  the  eclipse. — The  time  of  opposition  being  known,  that  of  the 
middle  of  the  eclipse  will  become  known  when  we  have  found  the  interval  (x)  em- 
ployed by  the  moon  in  passing  from  M  to  C'.  Now 

.  MC' 

(expressed  in  parts  of  an  hour)  x  =  — — ; 

h 

and  in   the   right-angled  triangle  CC'M  we  have  CC'  =  X,  and  <  C'CM  = 
<  C'N'C  =  I,  and  therefore  MC'  =  X  sin  I ;  whence,  by  substitution, 
X  sin  I        X  sin  I  X  sin  I  cos  I 


cos  I 

or,  (expressed  in  seconds,)  x  =  — _-!—_.  x  sin  I  .  .  (92). 


M — m 
Hence,  if  M  =  time  of  middle,  we  have 

3600s.  cos  1 


.  X  sin  I  ...  (93) 


M  — m 

It  is  obvious  that  the  upper  sign  is  to  be  used  when  the  latitude  is  increasing 
and  the  lower  sign  when  it  is  decreasing. 

The  distance  of  the  centre  of  the  moon  from  the  centre  of  the  shadow  at  the  mid 
die  of  the  eclipse, 

=  CM  =  CC'cos  C'CM  =  X  cos  I  .  .  .  (94). 

453.  Beginning  and  end  of  the  eclipse. — Let  any  point  I  of  the  relative  orbit  be 
the  place  of  the  moon's  centre  at  the  time  of  any  given  phase  of  the  eclipse.  Let 
t  =  the  interval  of  time  between  the  given  phase  and  the  middle ;  and  k  =  C/, 


CALCULATION  OF  A  LUNAR  ECLIPSE.  169 

the  distance  of  the  centres  of  the  moon  and  shadow.     In  the  interval  t  the  moon's 
centre  will  pass  over  the  distance  MZ  ;  hence 

M[=  M/.COS! 

h    ~~   M  —  m 


but,  MZ  =  \/c/2  —  clV?  =  V~W—tf  ct>S2  I  (equa.  94), 


C°S 


and  therefore  t  =  <J  k*  —  X2  cos2  I  ; 

M  —  m 


3600s.  cos  I 


or,  (in  seconds,)       t  =  """ua-"JS*  </  (fc  +  X  cos  I)  (A;  —A  cos  I) ...  (95) 
M  —  m 

Let  T'  denote  the  time  of  the  supposed  phase  of  the  eclipse,  and  M  the  time  of 
the  middle  ;  and  we  shall  have 

T'  =  M  -f  *,  or  T'  =  M  —  t , 

according  as  the  phase  follows  or  precedes  the  middle. 
Now,  at  the  beginning  and  end'of  the  eclipse,  we  have 

k=CforCf  = 

substituting  in  equation  (95)  we  obtain 
3600s.  cos  I 


-V  (s  +  d  -h  X  cos  I)  (s  -f  d  —  A  cos  I)  .  .  .  (96). 
JM  —  m 

t'  beir.g  found,  the  time  of  the  beginning  (B,;  and  the  time  of  the  end  (E,)  result 
from  the  equations 

B  =  M—  *',  E=M-H'. 

454.  Beginning  and  end  off-he  total  eclipse. — At  the  beginning  and  end  of  the 
total  eclipse,  k  =  Cg  =  Cg'  =  s  —  d  ;  whence,  by  equation  (95,) 


3600s.  cos  I 


t"  =      "  V  (s  —  d  -f  X  cos  I)  (s—d—  XcosI)  .  .  .(97): 

M  —  m 

and,  denoting  the  time  of  the  beginning  by  B'  and  the  time  of  the  end  by  E',  we 
have  B'  =  M  —  t",  E'  =  M  +  t". 

455.  Quantity  of  the  eclipse.  —  In  a  partial  eclipse  of  the  moon  the  magnitude 
or  quantity  of  the  eclipse  is  measured  by  the  relative  portion  of  that  diameter  of 
the  moon,  which,  if  produced,  would  pass  through  the  centre  of  the  earlh's  shad- 
ow,  that  is  involved  in  the  shadow.  The  whole  diameter  is  divided  into  twelve 
equal  parts,  called  Digits,  and  the  quantity  is  expressed  by  the  number  of  digits 
and  fractions  of  a  digit  in  the  part  immersed.  When  the  moon  passes  entirely 
within  the  shadow,  as  in  a  total  eclipse,  the  quantity'of  the  eclipse  is  expressed  by 
the  number  of  digits  contained  in  the  part  of  the  same  diameter  prolonged  outward, 
which  is  comprised  between  the  edge  of  the  shadow  and  the  inner  edge  of  the  moon. 
Thus  the  number  of  digits  contained  in  SN  (Fig.  74)  expresses  the  quantity  of  the 
eclipse  represented  in  the  figure.  Hence,  if  Q  =  the  quantity  of  the  eclipse,  we 
shall  have 

NS        12NS  _  12(NM  +  MS)       12  (NM  +  CS  —  CM) 
NV   =  NV  NV 

12  (d  -f  *  —  X  cos  I) 


If  X  cos  I  exceeds  (s  -{-  d)  there  will  be  no  eclipse.  If  it  is  intermediate  between 
(*  +  d}  and  (s  —  d)  there  will  be  a  partial  eclipse  ;  and  if  it  is  less  than  (*  —  d) 
the  eclipse  will  be  total. 

CONSTRUCTION  OF  AN  ECLIPSE  OF  THE  MOON. 

456.  The  times  of  the  different  phases  of  an  eclipse  of  the 
moon  may  easily  be  determined  by  a  geometrical  construction, 
within  a  minute  or  two  of  the  truth.  Draw  a  right  line  N'F 

22 


170 


ECLIPSES  OF  THE  SUN  AND  MOON. 


(Fig.  75)  to  represent  the  ecliptic ;  and  assume  upon  it  any 
point  C,  for  the  position  of  the  centre  of  the  earth's  shadow  at 
the  time  of  opposition.  Then,  having  fixed  upon  a  scale  of  equal 


parts,  lay  off  CR  =  M  —  m,  the  difference  of  the  hourly  motions 
of  the  sun  and  moon  in  longitude ;  and  draw  the  perpendiculars 
CC'  =  X  the  moon's  latitude  in  opposition,  and  RL'  =  X±  n,  the 
moon's  latitude  an  hour  after  opposition.  The  right  line  C'L', 
drawn  through  C'  and  L',  will  represent  the  moon's  relative  orbit. 
It  should  be  observed,  that  if  the  latitudes  are  south  they  must  be 
laid  off  below  N'F,  and  that  N'C'L'  will  be  inclined  to  the  right 
when  the  latitude  is  decreasing.  With  a  radius  CE  =s  (equation 
89)  describe  the  circle  EKFK',  which  will  represent  the  section 
of  the  earth's  shadow.  With  a  radius  =  s  +  d,  and  another  radius 
=  s  —  d,  describe  about  the  centre  C  arcs  intersecting  N'L'  in 
/,/',  and  g,  g' ;  /and/'  will  be  the  places  of  the  moon's  centre  at 
the  beginning  and  end  of  the  eclipse,  and  g  and  g'  the  places  at 
the  beginning  and  end  of  the  total  eclipse.  From  the  point  C  let 
fall  upon  N'C'L'  the  perpendicular  CM  ;  and  M  will  be  the  place 
of  the  moon's  centre  at  the  middle  of  the  eclipse.  To  render  the 
construction  explicit,  let  us  suppose  the  time  of  opposition  to  be 
7h.  23m.  15s.  At  this  time  the  moon's  centre  will  be  at  C'.  To 
find  its  place  at  7h.,  state  the  proportion,  60m. :  23m.  15s. :  :  moon's 
hourly  motion  on  the  relative  orbit  :  a  fourth  term.  This  fourth 
term  will  be  the  distance  of  the  moon's  centre  from  the  point  C'  at 
7  o'clock ;  and  if  it  be  taken  in  the  dividers  and  laid  off  on  the 
relative  orbit  from  C'  backward  to  the  point  7,  it  will  give  the 
moon's  place  at  that  hour.  This  being  found,  take  in  the  divi- 
ders the  moon's  hourly  motion  on  the  relative  orbit,  and  lay  it  off 
repeatedly,  both  forward  and  backward,  from  the  point  7,  and 
the  points  marked  off,  8,  9,  10,  6,  5,  will  be  the  moon's  places  at 
those  hours  respectively.  Now,  the  object  being  to  find  the  times 
at  which  the  moon's  centre  is  at  the  points/,/',  g,  g',  and  M,  let 


ECLIPSES  OF  THE  SUN LUMINOUS  FRUSTUM. 


in 


the  hour  spaces  thus  found  be  divided  into  quarters,  and  these 
subdivided  into  5-minute  or  minute  spaces,  and  the  times  answer- 
ing to  the  points  of  division  that  fall  nearest  to  these  points,  will 
be  within  a  minute  or  so  of  the  times  in  question.  For  example, 
the  point/'  falls  between  9  and  10,  and  thus  the  end  of  the  eclipse 
will  occur  somewhere  between  9  and  10  o'clock.  To  find  the  num- 
ber of  minutes  after  9  at  which  it  takes  place,  we  have  only  to 
divide  the  space  from  9  to  10  into  four  equal  parts  or  1 5-minute 
spaces,  subdivide  the  part  which  contains  /'  into  three  equal  parts 
or  5-minute  spaces,  and  again  that  one  of  these  smaller  parts 
within  which/7  lies,  into  five  equal  parts  or  minute  spaces.* 

ECLIPSES  OF  THE  SUN. 

457.  An  eclipse  of  the  sun  is  caused  by  the  interposition  of  the 
moon  between  the  sun  and  earth ;  whereby  the  whole,  or  part  of 
the  sun's  light,  is  prevented  from  falling  upon  certain  parts  of  the 
earth's  surface. 

Let  AGB  and  agb  (Fig.  76)  be  sections  of  the  sun  and  earth 


Fig.  76. 


by  a  plane  passing  through  their  centres  S  and  E,  A#,  BZ>  tan- 
gents to  the  circles  AGB  and  agb  on  the  same  side,  and  Ad,  Be 
tangents  to  the  same  on  opposite  sides.  The  figure  A06B  will  be 
a  section  through  the  axis,  of  a  frustum  of  a  cone  formed  by  rays 
tangent  to  the  sun  and  earth  on  the  same  side,  and  the  triangular 
space  Fed  will  be  a  section  of  a  cone  formed  by  rays  tangent  on 
opposite  sides.  An  eclipse  of  the  sun  will  take  place  somewhere 
upon  the  earth's  surface,  whenever  the  moon  comes  within  the 
frustum  AabB,  and  a  total  or  an  annular  eclipse  whenever  the 
moon  comes  within  the  cone  Fed. 

458.  Let  mm'M.  (Fig.  76)  be  a  circular  arc  described  about  the 
•  centre  E,  and  with  a  radius  equal  to  the  distance  of  the  centres 
of  the  moon  and  earth  at  the  time  of  conjunction.  The  angle 
?nES  is  the  apparent  semi-diameter  of  a  section  of  the  frustum, 
and  m'ES  the  apparent  semi-diameter  of  a  section  of  the  cone,  at 
the  distance  of  the  moon.  To  find  expressions  for  these  semi- 
diameters  in  terms  of  determinate  quantities,  let  the  first  be  de- 
noted by  A,  and  the  second  by  A' ;  and  let  P  =  the  parallax  of 


172  ECLIPSES  OF  THE  SUN  AND  MOON. 

the  moon,  p  =  the  parallax  of  the  sun;  and  £  =  the  semi-diameter 
of  the  sun.     Then  we  have 

mES  =  A  =  mEA.  +  AES  =  Ema  -  EAm  +  AES  ; 
or,  A  =  P  -  p  +  d  .  .  .  (99) ; 

and   m'ES  =  m'EE  -  BES  =  Em'c  -  EBm'  —  BES  ; 
or,  A'  =  P  —  p  -*  .  .  .  (100). 

Taking  the  mean  values  of  P,  p,  and  <$,  (441,)  we  find  for  the 
mean  value  of  A  1°  12'  53",  and  for  the  mean  value  of  A'  40'  51 ". 

459.  As  the  plane  of  the  moon's  orbit  is  not  coincident  with 
the  plane  of  the  ecliptic,  an  eclipse  of  the  sun  can  happen  only 
when  conjunction  or  new  moon  takes  place  in  one  of  the  nodes 
of  the  moon's  orbit,  or  so  near  it  that  the  moon's  latitude  does  not 
exceed  the  sum  of  the  semi-diameters  of  the  moon  and  of  the  lu- 
minous frustum  (457)  at  the  moon's  orbit.    This  may  be  illustrated 
by  means  of  Fig.  72,  already  used  for  a  lunar  eclipse,  by  supposing 
the  sun  to  be  in  the  directions  Es,  Es',  Es",  and  that  s,  s',  s",  are 
sections  of  the  luminous  frustum  corresponding  to  these  directions 
of  the  sun,  also  that  ra,  m-9  m",  represent  the  moon  in  the  cor- 
responding positions  of  conjunction.     Thus,  denoting  the  moon's 
semi-diameter  by  d,  and  the  greatest  latitude  of  the  moon  in  con- 
junction, at  which  an  eclipse  can  take  place,  by  L,  we  have 

L  =P  -p  +  d+d  .  .  .  (101). 

For  a  total  eclipse,  the  greatest  latitude  will  be  equal  to  the  sum 
of  the  semi-diameters  of  the  moon  and  the  luminous  cone.  Hence, 
denoting  it  by  L', 

L'  =  P-p-d  +  d  .  .  .  (102). 

In  order  that  an  annular  eclipse  may  take  place,  the  apparent 
s  emi-diameter  of  the  moon  must  be  less  than  that  of  the  sun,  and 
the  moon  must  come  at  conjunction  entirely  within  the  luminous 
frustum.  Whence,  if  L"  =  the  maximum  latitude  at  which  an 
annular  eclipse  is  possible,  we  have 

L"  =  P-p  +  8-d  .  .  .  (103). 

460.  In  the  same  manner  as  in  the  case  of  an  eclipse  of  the 
moon,  it  has  been  found  that  when  at  the  time  of  mean  new  moon 
the  difference  of  the  mean  longitudes  of  the  sun  or  moon  and  of 
the  node,  exceeds  19°  44',  there  cannot  be  an  eclipse  of  the  sun ; 
but  when  the  difference  is  less  than  13°  33',  there  must  be  one. 
These  numbers  are  called  the  Solar  Ecliptic  Limits. 

461.  In  order  to  discover  at  what  new  moons  in  the  course  of  a 
year  an  eclipse  of  the  sun  will  happen,  with  its  approximate  time, 
we  have  only  to  find  the  mean  longitudes  of  the  sun  and  node  at 
each  mean  new  moon  throughout  the  year,  (418,)  and  take  the 
difference  of  the  longitudes  and  compare  it  with  the  solar  ecliptic 
limits.     (For  a  more  direct  method  of  solving  this  problem,  see 
Prob.  XXVIII.) 

462.  Eclipses  both  of  the  sun  and  moon  recur  in  nearly  the 


NUilBEJl  OF  ECLIPSES  IN  A  YEAR.  173 

sail)'*  order  and  at  the  same-iutervals  at  the  expiration  of  a  period 
of  223  lunations,  or  18  years  of  365  days,  and  15  days;* which 
for  this  reason  is  called  the  Period  of  the  Eclipses.  For,  the 
time  of  a  revolution  of  the  sun  with  respect  to  the  moon's  node  is 
346. 6 1985 Id.,  and  the  time  of  a  synodic  revolution  of  the  moon 
is  29.5305887d.  These  numbers  are  very  nearly  in  the  ratio  of 
223  to  19.  Thus,  in  a  period  of  223  lunations,  the  sun  will  have 
returned  19  times  to  the  same  position  with  respect  to  the  moon's 
node,  and  at  the  expiration  of  this  period  will  be  in  the  same  posi- 
tion with  respect  to  the  moon  and  node  as  at  its  commencement. 
The  eclipses  which  occur  during  one  such  period  being  noted, 
subsequent  eclipses  are  easily  predicted. 

This  period  was  known  to  the  Chaldeans  and  Egyptians,  by 
whom  it  was  called  Saros. 

463.  As  the  solar  ecliptic  limits  are  more  extended  than  the  lu- 
nar, eclipses  of  the  sun  must  occur  more  frequently  than  eclipses 
of  the  moon. 

As  to  the  number  of  eclipses  of  both  luminaries,  there  cannot  be 
fewer  than  two  nor  more  than  seven  in  one  year.  The  most  usual 
number  is  four,  and  it  is  rare  to  have  more  than  six.  When  there 
are  seven  eclipses  in  a  year,  five  are  of  the  sun  and  two  of  the 
moon ;  and  when  but  two,  both  are  of  the  sun.  The  reason  is  ob- 
vious. The  sun  passes  by  both  nodes  of  the  moon's  orbit  but  once 
in  a  year,  unless  he  passes  by  one  of  them  in  the  beginning  of  the 
year,  in  which  case  he  will  pass  by  the  same  again  a  little  before 
the  end  of  the  year,  as  he  returns  to  the  same  node  in  a  period  of 
346  days.  Now,  if  the  sun  be  at  a  little  less  distance  than  19°  44' 
from  either  node  at  the  time  of  mean  new  moon,  he  may  be  eclipsed 
(4SO),  and  at  the  subsequent  opposition  the  moon  will  be  eclipsec 
near  the  other  node,  and  come  round  to  the  next  conjunction  before 
the  sun  is  13°  33'  from  the  former  node  :  and  when  three  eclipses 
happen  about  either  node,  the  like  number  commonly  happens 
about  the  opposite  one  ;  as  the  sun  comes  to  it  in  173  days  after 
wards,  and  six  lunations  contain  only  four  days  more.  Thus  there 
may  be  two  eclipses  of  the  sun  and  one  of  the  moon  about  each  of 
the  nodes  ;  and  the  twelfih  lunation  from  the  eclipse  in  the  begin 
ning  of  the  year  may  give  a  new  moon  before  the  year  is  ended, 
which,  in  consequence  of  the  retrogradation  of  the  nodes,  may  be 
within  the  solar  ecliptic  limit ;  and  hence  there  may  be  seven 
eclipses  in  a  year,  five  of  the  sun  and  two  of  the  moon.-  But  when 
the  moon  changes  in  either  of  the  nodes,  she  cannot  be  near  enough 
to  the  other  node,  at  the  next  full  moon,  to  be  eclipsed,  as  in  the 
interval  the  sun  will  move  over  an  arc  of  14°  32',  whereas  the 
greatest  lunar  ecliptic  limit  is  but  13°  21',  and  in  six  lunar  months 
afterwards  she  will  change  near  the  other  node  ;  in  this  case  there 
cannot  be  more  than  two  eclipses  in  a  year,  both  of  which  will  be 

*  More  exactly,  18  years  (of  365  days)  plus  15d  7h,  42m.  29s, 


174  ECLIPSES  OF  THE  SUN  AND  MOON. 

of  the  sun.  If  the  moon  changes  at  the  distance  of  a  few  degrees 
from  either  node,  then  an  eclipse  both  of  the  sun  and  moon  will 
probably  occur  in  the  passage  of  that  node  and  also  of  the  other. 

464.  Although  solar  eclipses  are  more  frequent  than  lunar,  when 
considered  with  respect  to  the  whole  earth,  yet  at  any  given  place 
more  lunar  than  solar  eclipses  are  seen.     The  reason  of  this  cir- 
cumstance is,  that  an  eclipse  of  the  sun  (unlike  an  eclipse  of  the 
moon)  is  visible  only  over  a  part  of  a  hemisphere  of  the  earth.    To 
show  this,  suppose  two  lines  to  be  drawn  from  the  centre  of  the 
moon  tangent  to  the  earth  at  opposite  points  :  they  will  make  an 
angle  with  each  other  equal  to  double  the  moon's  horizontal  paral- 
lax, or  of  1°  54'.     Therefore,  should  an  observer  situated  at  one 
of  the  points  of  tangency,  refer  the  centre  of  the  moon  to  the  cen- 
tre of  the  sun,  an  observer  at  the  other  would  see  the  centres  of 
these  bodies  distant  from  each  other  at  an  angle  of  1°  54',  and  their 
nearest  limbs  separated  by  an  arc  of  more  than  1°. 

465.  Instead  of  regarding  an  eclipse  of  the  sun  as  produced  by 
an  interposition  of  the  moon  between  the  sun  and  earth,  as  we  have 
hitherto  considered  it,  we  may  regard  it  as  occasioned  by  the  moon's 
shadow  falling  upon  the  earth.     Fig.  77  represents  the   moon's 
shadow,  as  projected  from  the  sun  and  covering  a  portion  of  the 
earth's  surface.    Wherever  the  umbra  falls,  there  is  a  total  eclipse  ; 
and  wherever  the  penumbra  falls,  a  partial  eclipse. 

Fig.  77. 


466.  In  order  to  discover  the  extent  of  the  portion  of  the  earth's 
surface  over  which  the  eclipse  is  visible  at  any  particular  time, 
we  have  only  to  find  the  breadth  of  the  portion  of  the  earth  covered 
by  the  penumbral  shadow  of  the  moon ;  but  we  will  first  ascertain 
the  length  of  the  moon's  shadow.  As  seen  at  the  vertex  of  the 
moon's  shadow,  the  apparent  diameters  of  the  moon  and  sun  are 
equal.  Now,  as  seen  at  the  centre  of  the  earth,  they  are  nearly 
equal,  sometimes  the  one  being  a  little  greater  and  sometimes  the 
other.  It  follows,  therefore,  that  the  length  of  the  moon's  shadow 
is  about  equal  to  the  distance  of  the  earthy  being  sometimes  a  little 
greater  and  at  other  times  a  little  less. 


175 


afey  the 


ECLIPSES  OF  THE  SUN. — MOON  SiPH^DQW.      t> 

When  the  apparent  diameter  of  the  moon  is  tffespi 
shadow  will  extend  beyond  the  earth's  centre  ;  and  wfitt^  the  -ap- 
parent diameter  of  the  sun  is  the  greater,  it  will  fall  short'  o/^t.  If/  • 
we  increase  the  mean  apparent  diameter  of  the  moon  as  seen'from  ^ 
the  earth's  centre,  viz.  31' 7",  by  ^>  the  ratio  of  the  radius  ortfie 
earth  to  the  distance  of  the  moon,  we  shall  have  31'  38"  far  th^ 
mean  apparent  diameter  of  the  moon  as  seen  from  the  nearest  point 
of  the  earth's  surface.  Comparing  this  with  the  mean  apparent 
diameter  of  the  sun  as  viewed  from  the  same  point,  which  is  sen- 
sibly the  same  as  at  the  centre  of  the  earth,  or  32'  2",  we  perceive 
that  it  is  less ;  from  which  we  conclude,  that  when  the  sun  and 
moon  are  each  at  their  mean  distance  from  the  earth,  the  shadow 
of  the  moon  does  not  extend  as  far  as  the  earth's  surface. 

467.  To  find  a  general  expression  for  the  length  of  the  moon's 
shadow,  let  AGB,  a'g'b',  and  agb  (Fig.  78)  be  sections  of  the  sun, 

Fig.  78. 


moon,  and  earth,  by  a  plane  passing  through  their  centres  S,  M, 
and  E,  supposed  to  be  in  the  same  right  line,  and  Aa',  Bb'  tan- 
gents to  the  circles  AGB,  a'g'b' :  then  a'K6'  will  represent  the 
moon's  shadow.  Let  L  ==  the  length  of  the  shadow  ;  D  =  the  dis- 
tance of  the  mocn  ;  D'  =  the  distance  of  the  sun  ;  d  =  the  appa- 
rent semi-diameter  of  the  moon ;  and  5  =  apparent  semi-diameter 
of  the  sun.  At  K  the  vertex  of  the  shadow,  MKa'  the  apparent 
semi-diameter  of  the  moon,  will  be  equal  to  SKA  the  apparent  se- 
mi-diameter of  the  sun ;  and  as  the  distance  of  this  point  from  the 
centre  of  the  earth,  even  when  it  is  the  greatest,  is  small  in  com- 
parison with  the  distance  of  the  sun  (466),  the  apparent  semi-diam- 
eter of  the  sun  will  always  be  very  nearly  the  same  to  an  observer 
situated  at  K  as  to  one  situated  at  the  centre  of  the  earth.  Now, 
since  the  apparent  semi-diameter  of  the  moon  is  inversely  propor- 
tional to  its  distance, 

angle  MKa' :  d  :  :  ME  :  MK  ; 
and  thus,  3  :  d  : :  ME  :  MK  :  :  D  :  L  (nearly) : 

whence,  L=D-|  .  .  .  (104). 

If  a  more  accurate  result  be  desired,  we  have  only  to  repeat  the  cal- 
culations, after  having  diminished  5  in  the  ratio  of  D'  to  (D'-fL— D). 

468.  Now,  to  find  the  breadth  of  the  portion  of  the  earth's  surface  covered  by 
the  penumbral  shadow,  let  the  lines  Ad',  Be'  (Fig.  78)  be  drawn  tangent  to  the 
circles  AGB,  a'g'b',  on  opposite  sides,  aud  prolonged  on  to  the  earth.  The  space 


176  ECLIPSES  OF  THE  SUN  AND  MOON. 

hc'd'k  will  represent  the  penumbra  of  the  moon's  shadow,  and  the  arc  gJi  one  half 
the  breadth  of  the  portion  of  the  earth's  surface  covered  by  it.  Let  this  arc  or  the 
angle  g-EA  =  S,  and  denote  the  semi-diameter  of  the  sun  and  the  semi-diameter 
and  parallax  of  the  moon  by  the  same  letters  as  in  previous  articles.  The  triangle 
MEA  gives 

angle  MEA  =  S  =  MAZ  —  AME. 

The  angle  AME  is  the  moon's  parallax  in  altitude  at  the  station  A,  and  MAZ  is 
its  zenith  distance  at  the  same  station.  Denote  the  former  by  P'  and  the  latter  by  Z. 
Thus,  S  =  Z  —  F  .  .  .  (105). 

The  triangle  AMS  gives 

AME  =  F=MSA+MAS; 

MAS  =  d-\-t> ;  and  MSA  is  the  sun's  parallax  in  altitude  at  the  station  A:  let  it 
be  denoted  by  p'.  We  have,  then, 

F  =  d  +  S  +p'  =  d  +  6  (nearly)  .  .  .  (106); 
and  to  find  Z  we  have  (equa.  9,  p.  51), 

F  =  P  sin  Z,  or  sin  Z  =  -  .  .  .  (107). 

F  and  Z  being  found  by  these  equations,  equa.  (105)  will  then  make  known  the 
value  of  S. 

If  great  accuracy  is  required,  the  calculation  must  be  repeated,  giving  now  to 
p'  in  equation  (106)  the  value  furnished  by  equation  (9)  which  expresses  the  rela- 
tion between  the  parallax  in  altitude  of  a  body  and  its  horizontal  parallax,  instead 
of  neglecting  it  as  before  ;  and  Z  must  be  computed  from  the  following  equation : 

£*.. 

sin  P 

The  penumbral  shadow  will  obviously  attain  to  its  greatest 
breadth  when  the  sun  is  in  its  perigee  and  the  moon  is  in  its  apo- 
gee. The  values  of  d,  <5,  and  P  under  these  circumstances  are  re- 
spectively 14'  41",  16'  18'',  and  53'  48".  Performing  the  calcula- 
tions, we  find  that  the  breadth  of  the  greatest  portion  of  the  earths 
surface  ever  covered  by  the  penumbral  shadow  is  69°  18',  or  about 
4800  miles. 

469.  The  breadth  of  the  spot  comprehended  within  the  umbra 
may  be  found  in  a  similar  manner. 

The  arc  gh'  (Fig.  78)  represents  one  half  of  it :  denote  this  arc  or  the  equal  an- 
gle ffEA'  by  S'. 

MEA'  ==  S'  =  MA'Z'  —  A'ME ; 
or,  S'  =  Z-F  .  .  .  (109). 

A'ME  =  F  =  MSA'  +  MA'S; 
but  MA'S  =  d  —  5,  and  MSA'  =  p',  sun's  parallax  in  altitude  at  A' ;  whence, 

Y'  =  d  —  $+p'  =  d  —  S  (nearly)  .  .  .  (110): 
and  we  ha^«,  as  before, 

F  =  PsinZ,orsinZ=—  .  .  .  (111). 

The  greatest  breadth  will  obtain  when  the  sun  is  in  its  apogee 
and  the  moon  is  in  its  perigee.  We  shall  then  have 

<5  =  15'  45",  d  =  16'  45",  P  =  61'  24". 

Making  use  of  these  numbers,  we  deduce  for  the  maximum 
breadth  of  the  portion  of  the  earths  surface  covered  by  the  moon's 
shadow,  1°  50',  or  127  miles. 

470.  It  should  be  observed  that  the  deductions  of  the  last  two 


CALCULATION  OF  AN  ECLIPSE  OF  THE  SUN.  177 

articles  answer  to  the  supposition  that  the  moon  is  in  the  node,  and 
that  the  axis  of  the  shadow  and  penumbra  passes  through  the  cen- 
tre of  the  earth.  In  every  other  case,  both  the  shadow  and  pe- 
numbra will  be  cut  obliquely  by  the  earth's  surface,  and  the  sec- 
tions will  be  ovals,  and  very  nearly  true  ellipses,  the  lengths  of 
which  may  materially  exceed  the  above  determinations. 

471.  Parallax  not  only  causes  the  eclipse  to  be  visible  at  some 
places  and  invisible  at  others,  as  shown  in  Art.  464  ;  but,  by  making 
the  distance  of  the  centres  of  the  sun  and  moon  unequal,  renders 
the  circumstances  of  the  eclipse  at  those  places  where  it  is  visible 
different  at  each  place.  This  may  also  be  inferred  from  the  cir- 
cumstance that  the  different  places,  covered  at  any  time  by  the 
shadow  of  the  moon,  will  be  differently  situated  within  this  shadow. 
It  will  be  seen,  therefore,  that  an  eclipse  of  the  sun  has  to  be  con- 
sidered in  two  points  of  view:  1st.  With  respect  to  the  whole 
earth,  or  as  a  general  eclipse ;  and,  2d.  With  respect  to  a  particu- 
lar place. 

472.  The  following  are  the  principal  facts  relative  to  eclipses  of  the  sun  that 
remain  to  be  noticed :  1st.  The  duration  of  a  general  eclipse  of  the  sun  cannot  ex- 
ceed about  6  hours.     2d.  A  solar  eclipse  does  not  happen  at  the  same  time  at  all 
places  where  it  is  seen :  as  the  motion  of  the  moon  beyond  the  sun,  and  conse- 
quently of  its  shadow,  is  from  west  to  east,  the  eclipse  must  begin  earlier  at  the 
western  parts  and  later  at  the  eastern.     3d.  The  moon's  shadow,  being  tangent  to 
the  earth  at  the  commencement  and  end  of  the  eclipse,  the  sun  will  be  just  rising 
at  the  place  where  the  eclipse  is  first  seen,  and  just  setting  at  the  place  where  it  is 
last  seen.  At  the  intermediate  places,  the  sun  will  at  the  time  of  the  beginning  and 
end  of  the  eclipse  have  various  altitudes.     4th.  An  eclipse  of  the  sun  begins  on  the 
western  side  and  ends  on  the  eastern.  5th.  When  the  straight  line  passing  through 
the  centres  of  the  sun  and  moon  passes  also  through  the  place  of  the  spectator,. the 
eclipse  is  said  to  be  central:  a  central  eclipse  may  be  either  annular  or  total,  ac- 
cording as  the  apparent  diameter  of  the  sun  is  greater  than  that  of  the  moon,  or 
the  reverse.  6th.  A  total  eclipse  of  the  sun  cannot  last  at  any  one  place  more  than 
eight  minutes  ;  and  an  annular  eclipse  more  than  twelve  and  a  half  minutes.    7th. 
In  most  solar  eclipses  the  moon's  disc  is  covered  with  a  faint  light,  a  phenomenon 
which  is  attributed  to  the  reflection  of  the  light  from  the  illuminated  part  of  the 
•arth. 

CALCULATION    OF    AN    ECLIPSE    OF    THE    SUN. 

(1.)  Of  the  circumstances  of  th,e  general  eclipse. 

473.  It  is  a  simple  inference  from  what  has  been  established  in  Art.. 459,  that  an 
eclipse  of  the  sun  will  begin  and  end  upon  the  earth,  at  the  times  before  and  after 
conjunction,  when  the  distance  of  the   centres  of  the  moon  and  sun  is  equal  to 
P — p-\-S-{-d't  that  the  total  eclipse  will  begin  and  end  when  this  distance  is 
equal  to  P — p  —  &-\-d\  and  the  annular  eclipse  when  the  distance  is  equal  to 
P—  p  +  &^td. 

474.  The  times  of  the  various  phases  of  the  general  eclipse  of  the  sun  may  bo 
obtained  by  a  process  precisely  analogous  to  that  by  which  the  times  of  the  pknses 
of  an  eclipse  of  the  moon  are  found.    Let  C  (Fig.  79)  be  the  centre  of  the  sun,  and 
C'  the  centre  of  the  moon,  at  the  time  of  conjunction.     We  may  suppose  the  sun 
to  remain  stationary  at  C,  if  we  attribute  to  the  moon  a  motion  equal  to  its  mo- 
tion relative  to  the  sun ;  for,  on  this  supposition,  the  distance  of  the  centres  of  the 
two  bodies  will,  at  any  given  period  during  the  eclipse,  be  the  same  as  that  which 
obtains  in  the  actual  state  of  the  case.     Let  N'C'L'  represent  the  orbit  that  would 
be  described  by  the  moon  if  it  had  such  a  motion,  which  is  called  the  Relative  Or- 
bit.    Let  CM  be  drawn  perpendicular  to  it ;  and  let  C/=  C/  =  P—  ^-4-3-f  dt 
and  Cg  =  Cg'  =P  —  p  —  S  +  d,  orP  — p-\-&  —  dt  according  as  the  eclipse  is  to- 

23 


178 


ECLIPSES  OF  THE  SUN  AND  MOON. 


tal  or  annular.  Then,  M  will  be  the  place  of  the  moon's  centre  at  the  middle  of 
the  eclipse  ;  /and/  the  places  at  the  beginning  and  end  of  the  eclipse  ;  a.ndg  and 
g'  the  places  at  the  beginning  and  end  of  the  total,  or  of  the  annular  eclipse.  We 
•hall  thus  have,  as  in  eclipses  of  the  moon, 

Fig.  79. 


tan?  I  =  -=5 ,  CM  =  X  cos  I,  C'M  =  X  sin  I  ...  (112). 

J*L — m 

3600s.  X  sin  I  cos  I 


Jutervalfrom  con.  to  mid.    _ ,_ 

ivi  — m. 

Interval  from  middle  to  beginning  or  end 

3600s.  cos  I 

:~lK  —  m 
Interval  for  total  eclipse 

3600s.  cos  I 


.  .  .  (113). 


—  XcosI)    .  .  .  (114). 


=  "u:r  °"°  V(*"  -f  X  cos  I)  (k»  -  X  cos  I) 

IVL  ™~  fTt 


(115). 


Interval  for  annular  eclipse 
3600s.  cos  I 


M  —  m 

Quantity  = 


"+X  cos  I)  (A:'"  — X  cos  I)  ...  (116). 
—  X  cos  I) 


..  .  (117). 


—d  .  .  .  (118). 

The  letters  X,  M,  m,  &c.,  represent  quantities  of  the  same  name  as  in  the  formulas 
for  a  lunar  eclipse  ;  but  they  designate  the  values  of  these  quantities  at  the  time  of 
conjunction,  instead  of  opposition.  These  values  are  in  practice  obtained  from  ta- 
bles of  the  sun  and  moon,  as  in  a  lunar  eclipse. 

475.  The  times  of  the  different  circumstances  of  a  general  eclipse  of  the  sun 
may  also  be  found  within  a  minute  or  two  of  the  truth,  by  construction,  in  a  pre- 
cisely similar  manner  with  those  of  an  eclipse  of  the  moon,  (456.) 

(2.)   Of  the  phases  of  the  eclipse  at  a  particular  place. 

476.  The  phase  of  the  eclipse,  which  obtains  at  any  instant  at  a  given  place,  is 
indicated  by  the  relation  between  the  apparent  distance  of  the  centres  of  the  sun 
and  moon,  and  the  sum,  or  difference,  of  their  apparent  semi-diameters  :  and  the 
calculation  of  the  time  of  any  given  phase  of  the  eclipse,  consists  in  the  calculation 
of  the  time  when  the  apparent  distance  of  the  centres  has  the  value  relative  to  the 
sum  or  difference  of  the  semi-diameters,  answering  to  the  given  phase,     ihus,  if 
we  wish  to  find  the  time  of  the  beginning  of  the  eclipse,  we  have  to  seek  the  time 
when  the  apparent  distance  of  the  centres  of  the  sun  and  moon  first  becomes  equal 
to  the  sum  of  their  apparent  semi-diameters. 

477.  The  calculation  of  the  different  phases  of  an  eclipse  of  the  sun,  for  a  par- 
ticular place,  involves,  then,  the  determination  of  the  apparent  distance  of  the  cen- 
tres of  the  sun  and  moon,  and  of  the  apparent  semi-diameters  .of  the  two  bodies, 
at  certain  stated  periods. 

The  true  semi-diameter  of  the  sun,  as  given  by  the  tables,  may  be  taken  for  the 
apparent  without  material  error.  For  the  method  of  computing  the  apparent  semi- 
diameter  of  the  moon,  for  any  given  time  and  place,  see  Problem  XVII. 


SOLAR  ECLIPSE. APPROXIMATE  TIMES  OF  PHASES.     179 

478.  According  to  the  celebrated  astronomer  Dus6jour,  in  order  to  make  the  ob- 
servations agree  with  theory,  it  is  necessary  to  diminish  the  sun's  semi -diameter,  as 
it  is  given  by  the  tables,  3".5.     This  circumstance  is  explained  by  supposing  that 
the  apparent  diameter  of  the  sun  is  amplified,  by  reason  of  the  very  lively  impres- 
sion wlu'ch  its  light  makes  upon  the  eye.     This  amplification  is  called  Irradiation. 
He  also  thinks  that  the  semi-diameter  of  the  moon  ought  to  be  diminished  2",  to 
make  allowance  for  an  Inflexion  of  the  light  which  passes  near  the  border  of  this 
luminary,  supposed  to  be  produced  by  its  atmosphere.     It  must  be  observed,  how- 
ever, that  the  astronomers  of  the  present  day  do  not  agree  either  as  to  the  neces- 
sity or  the  amount  of  the  diminutions  just  spoken  of. 

479.  The  determination  of  the  apparent  distance  of  the  centres  of  the  sun  and 
moon  may  easily  be  accomplished,  as  will  be  shown  in  the  sequel,  when  the  ap- 
parent longitude  and  latitude  of  the  two  bodies  have  been  found.     Now,  the  true 
longitude  of  the  sun,  and  the  true  longitude  and  latitude  of  the  moon,  may  be  found 
from  the  tables,  (Probs.  IX  and  XIV) ;  and  from  these  the  apparent  longitudes  and 
latitudes  may  be  deduced  by  correcting  for  the  parallax.     But  the  customary  mode 
of  proceeding  is  a  little  different  from  this :  the  true  ivm  jitude  and  latitude  of  the 
sun  are  employed  instead  of  the  apparent,  and  the  parallax  of  the  sun  is  referred  to 
the  moon  ;  that  is,  the  difference  between  the  parallax  of  the  moon  and  that  of  the 
sun  is,  by  fiction,  taken  as  the  parallax  of  the  moon.  This  supposititious  parallax  is 
called  the  moon's  Relative  Parallax.     (See  Prob.  XVII.) 

480.  We  will  first  show  how  to  find  the  approximate  times  of  the  different  phases 
of  the  eclipse.     Put  T  =  the  time  of  new  moon,  known  to  within  5  or  10  minutes. 
(Prob.  XXVII.)    For  the  time  T  calculate  by  the  tables  the  sun's  longitude,  hourly 
motion,  and  semi-diameter,  and  the  moon's  longitude,  latitude,  horizontal  parallax, 
semi-diameter,  and  hourly  motions  in  longitude  and  latitude.     Subtract  the  sun's 
horizontal  parallax  from  the  reduced  horizontal  parallax  of  the  moon,*  and  calcu- 
late the  apparent  longitude  and  latitude,  and  the  apparent  semi-diameter  of  the 
moon.     From  a  comparison  of  the  apparent  longitude  of  the  moon  with  the  true 
longitude  of  the  sun,  we  shall  know  whether  apparent  ecliptic  conjunction  occurs 
before  or  after  the  time  T.     Let  T'  denote  the  time  an  hour  earlier  or  later  than 
the  time  T,  according  as  the  apparent  conjunction  is  earlier  or  later.     With  the 
sun  and  moon's  longitudes,  the  moon's  latitude,  and  the  hourly  motions  in  longi- 
tude and  latitude,  at  the  time  T,  calculate  the  longitudes  and  the  moon's  latitude 
for  the  time  T' ;  and  for  this  time  also  calculate  the  moon's  apparent  longitude 
and  latitude.     Take  the  difference  between  the  apparent  longitude  of  the  moon  and 
the  true  longitude  of  the  sun  at  the  time  T,  and  it  will  be  the  apparent  distance 
of  the  moon  from  the  sun  in  longitude,  at  this  time.     Let  it  be  denoted  by  n.  Find, 
in  like  manner,  the  apparent  distance  of  the  moon  from  the  sun  in  longitude  at  the 
time  T',  and  denote  it  by  n'.  In  the  same  manner  as  at  the  time  T,  we  find  wheth- 
er apparent  conjunction  occurs  before  or  after  the  tinieT'.     If  it  occurs  between 
the  times  T  and  T',  the  sum  of  n  and  n',  otherwise  their  difference,  will  be  the 
apparent  relative  motion  of  the  sun  and  moon  in  longitude  in  the  interval  T'  —  T, 
or  T  —  T'  ;  from  which  the  relative  hourly  motion  will  become  known.     The  dif- 
ference of  the  apparent  latitudes  of  the  moon,  at  the  times  T  and  T',  will  make 
known  the  apparent  relative  hourly  motion  in  latitude.     With  the  relative  hourly 
motion  in  longitude  and  the  difference  of  the  apparent  longitudes  at  the  time  T, 
find  by  simple  proportion  the  interval  between  the  time  T  and  the  time  of  apparent 
ecliptic  conjunction  ;  and  then,  with  the  apparent  latitude  of  the  moon  at  the  time 
T  and  its  hourly  motion  in  latitude,  find  the  apparent  latitude  at  the  time  of  ap- 
parent conjunction  thus  determined.     Then,  knowing  the  relative  hourly  motion 
of  the  sun  and  moon  in  longitude  and  latitude,  together  with  the  time  of  apparent 
conjunction,  and  the  apparent  latitude  at  that  time,  and  regarding1  the  apparent 
relative  orbit  of  the  moon  as  a  right  line,  (which  it  is  nearly,)  it  is  plain  that  the 
time  of  beginning,  greatest  obscuration,  and  end,  as  well  as  the  quantity  of  the 
eclipse,  may  be  calculated  after  the  same  manner  as  in  the  general  eclipse  ;  the 
disc  of  the  sun  answering  to  the  section  of  the  luminous  frustum  mentioned  in  Art 


*  The  reduced  horizontal  parallax  of  the  moon  is  its  horizontal  parallax  as  re  • 
duced  from  the  equator  to  the  given  place.  (See  Prob.  XV.) 


180 


ECLIPSES  OF  THE  SUN  AND  MOON. 


457,  and  the  apparent  elements 
answering  to  the  true.  Let  C 
(Fig.  80)  represent,  the  centre 
of  the  sun  supposed  stationary, 
CO'  the  apparent  latitude  of  the 
moon  at  apparent  conjunction. 
N'C'  the  apparent  relative  orbit 
of  the  moon,  determined  by  its 
passing  through  the  point  C' 
and  making  a  determinate  an- 
gle with  the  ecliptic  NT,  or  by- 
its  passing  through  the  situa- 
tions of  the  moon  at  the  times 
T  and  T'.  Also,  let  RKFK' 
be  the  border  of  the  sun's  disc ; 
f,f  the  positions  of  the  moon's 

centre  at  the  beginning  aud  end  of  the  eclipse,  determined  by  describing  a  circle 

around  C  as  a  centre,  with  a  radius  equal  to  the  sum  of  the  apparent  semi-diame. 

ters  of  the  sun  and  moon  ;  and  M  (the  foot  of  the  perpendicular  let  fall  from  C 

upon  N'C')  its  position  at  the  time  of  greatest  obscuration. 

If  the  eclipse  should  be  total  or  annular,  then  g,  g'  will  be  the  positions  of  the 

moon's  centre  at  the   beginning  and  end  of  the  total  or  annular  eclipse  ;  these 

points  being  determined  by  describing  a  circle  around  C  as  a  centre,  and  with  a 

radius   equal  to  the  difference   of  the   apparent  semi-diameters  of  the  sun  and 

moon. 

The  results  will  be  a  closer  approximation  to  the  truth,  if  the  same  calculations 

that  are  made  for  the  time  T'  be  made  also  for  another  time  T". 

The  various  circumstances  of  the  eclipse  may  also  be  had  by  construction,  after 

the  same  manner  as  in  a  lunar  eclipse,  (456.) 

481.  In  order  to  be  able  to  observe  the  beginning  or  end  of  a  solar  eclipse,  it  is 
necessary  to  know  the  position  of  the  point  on  the  sun's  limb  where  the  first  or 
last  contact  takes  place.     The  situation  of  these  points  is  designated  by  the  dis- 
tance on  the  limb,  intercepted  between  them  and  the  highest  point  of  the  limb,  call- 
ed the  Vertex.     The  contacts  will  take  place  at  the  points  t,  t' ,  (Fig.  80,)  on  the 
lines  C/,  Cf.     To  find  the  position  of  the  vertex,  with  the  sun's  longitude  found 
for  the  beginning  of  the  eclipse,  calculate  the  angle  of  position  of  the  sun  at  that 
time,  (see  Prob.  XIII,)  and  lay  it  off  to  the  right  of  the  circle  of  latitude  CK  when 
the  sun's  longitude  is  between  90°  and  270°,  and  to  the  left  when  the  longitude  is 
less  than  90°  or  more  than  270°.     Suppose  CP  to  be  the  circle  of  declination  thus 
determined.     Next,  let  Z  (Fig.  24,  p.  47)  be  the  zenith,  P  the  xelevated  pole,  and  S 
the  sun  ;  then  in  the  triangle  ZPS  we  shall  know  ZP  the  co-latitude,  ZPS  the  hour 
angle  of  the  sun,  and  we  may  deduce  PS,  the  co-declination  of  the  sun,  from  the 
longitude  of  the  sun  as  derived  from  the  tables,  (equa.  35.)     These  three  quanti- 
ties being  known,  ZSP,  the  angle  made  by  the  vertical  through  the  sun  with  its 
circle  of  declination,  may  be  computed ;  and  being  laid  off  in  the  figure  to  the 
right  or  left  of  CP,  (Fig.  80,)  according  as  the  time  of  beginning  is  before  or  after 
noon,  the  point  Z  or  Z',  as  the  case  may  be,  in  which  the  vertical  intersects  the 
limb  RKK',  will  be  the  vertex,  and  the  arc  Z<,  or  Z't,  on  the  limb,  will  ascertain 
the  situation  of  t,  the  first  point  of  contact,  with  respect  to  it. 

The  situation  of  the  last  point  of  contact  may  be  found  by  the  same  mode  of 
proceeding. 

482.  Let  us  now  show  how  to  find  the  exact  times  of  the  beginning,  greatest 
obscuration,  and  end  of  the  eclipse,  the  approximate  times  being  known.     Let  B 
designate  the  approximate  time  of  beginning,  taken  to  the  nearest  minute.     Cal- 
culate for  the  time  B  by  means  of  the  tables,  the  sun's  longitude,  hourly  motion, 
and  semi-diameter ;  also  the  moon's  longitude,  latitude,  horizontal  parallax,  semi- 
diameter,  and  hourly  motions  in  longitude  and  latitude.     Then,  making  use  of  thc« 
relative  parallax,  calculate  the  apparent  longitude,  latitude,  and  semi-diameter  of 
the  moon.     Subtract  the  apparent  longitude  of  the  moon  from  the  true  longitude 
of  the  sun  ;  the  difference  will  be  the  apparent  distance  of  the  moon  from  the  gun 
in  longitude  :  let  it  be  denoted  by  a.     Denote  the  apparent  latitude  of  the  moon 
byX. 


SOLAR  ECLIPSE TRUE  TIMES  OF  PHASES.  181 

Now,  let  EC  (Fig  81)  represent  an  arc  of  the  ecliptic, 
and  K  its  pole ;  and  let  S  be  the  situation  of  the  sun, 
and  M  the  apparent  situation  of  the  moon  at  the  time  B. 
Then  MS  is  the  apparent  distance  of  the  centres  of  the 
two  bodies  at  this  time.  Denote  it  by  A.  Sm  =  a, 
and  Mm  =  A.  The  right-angled  triangle  MSra  being 
very  small,  may  be  considered  as  a  plane  triangle,  and 
we  therefore  have,  to  determine  A,  the  equation 
A2  =  a2-r-A2  .  .  .  (119).* 

483.  Having  computed  the  value  of  A,  we  find,  by 
comparing  it  with  the  sum  of  the  apparent  semi-diame- 
ters of  the  sun  and  moon,  whether  the  beginning  of  the 
eclipse  occurs  before  or  after  the  approximate  time  B.  Fix 
upon  a  time  some  4  or  5  minutes  before  or  after  B,  ac- 
cording as  the  beginning  is  before  or  after,  and  call  it  B'. 
With  the  sun  and  moon's  longitudes,  the  moon's  latitude,  and  the  hourly  motions 
in  longitude  and  latitude,  at  the  time  B,  find  the  longitudes  and  the  moon's  lati- 
tude at  the  time  B',  and  compute  for  this  tinae  thft  apparent  longitude,  latitude, 
and  semi-diameter  of  the  moon.  Subtract  the  apparent  longitude  of  the  moon 
from  the  true  longitude  of  the  sun,  and  we  shall  have  the  apparent  distance  of  the 
moon  from  the  sun  at  the  time  B'.  Take  the  difference  between  this  and  the  same 
distance  a  at  the-time  B,  and  we  shall  have  the  apparent  relative  motion  of  the 
sun  and  moon  in  longitude  during  the  interval  of  time  between  B  and  B'.  Then 
find,  by  simple  proportion,  the  apparent  relative  hourly  motion  in  longitude,  and 
denote  it  by  k.  Take  the  difference  between  the  apparent  latitudes  of  the  moon 
at  the  times  B  and  B',  and  it  will  be  the  apparent  relative  motion  of  the  sun  and 
moon  in  latitude,  in  the  interval;  from  which  deduce  the  apparent  relative  hourly 
motion  in  latitude,  and  call  it  n.  Now,  put  t  =  the  interval  between  the  ap- 
proximate and  true  times  of  the  beginning  of  the  eclipse,  and  suppose  S  and  M 
(Fig.  81)  to  be  the  situations  of  the  sun  and  moon  at  the  true  time  of  beginning. 
In  the  time  /.  the  apparent  relative  motions  in  longitude  and  latitude  will  be,  re- 
spectively, equal  to  kt  and  nt,  and  accordingly  we  shall  have 

Sm  =  a  —  kt,  MOT  =  A  -|-  nt. 

The  small  right-angled  triangle  SMm  may  be  considered  as  a  plane  triangle  ;  the 
hypothenuse  SM  =  i//  ==  the  sum  of  the  apparent  semi-diameters  of  the  sun  and 
moon,  minus  5".5,  (478.)  We  have  then,  to  find  t,  the  equation 


or,  developing  and  transposing, 

(n2_|_  jfc2)  *2_  2  (ak  —  An)  *  =  ^2_  (a2  +  X2)  =  i//2—  A2  J 
making  A  =  ^  —  A2,  and  B  =  ak  —  An,  (n2  -f  &2)  t2  _  2B*  =  A, 


-  .  .  .  (120, 


The  negative  sign  must  be  prefixed  to  the  radical,  for,  if  we  suppose  A  to  be  equal 
to  zero,  t  must  be  equal  to  zero.  Multiplying  the  numerator  and  denominator  by 
B-f.  V  B2-|-A  (na  +  A8),  and  restoring  the  value  of  A,  we  obtain 

3600s.  (A2— 
(in  seconds)  t= 


Although  this  equation  has  been  investigated  for  the  beginning  of  the  eclipse,  it 
is  plain  that  it  will  answer  equally  well  for  the  determination  of  the  other  phases, 

*  In  place  of  equation  (119)  the  following  equations  may  be  employed  in  loga- 
hmic computation: 


j _      _-j — 

rithmic  computation 
where  6  is  an  auxiliary  arc 


182  ECJJPSES  OP  THE  SUN  AND  MOON. 

if  we  give  the  proper  values  and  signs  to  ^,  a,  A,  n,  and  k.     k  is  positive  before 
conjunction  :iml  negative  after  it,  and  the  radical  quantity  is  negative  after  con- 
junction ;  n  \3  negative,  when  the  moon  appears  to  recede  from  the  north  pole  of 
the  ecliptic  ;  A  hr\£  the  sign  — ,  when  it  is  south  ;  a  is  always  positive.* 
The  value  of  t  taken  with  its  sign  is  to  be  added  to  the  time  B. 

484.  The  values  of  the  quantities  a,  A,  n,  and  k,  are  found  for  the  other  phases 
after  the  same  manner  a*  for  the  beginning. 

To  obtain  the  value  of  ^  at  the  time  of  greatest  obscuration,  find  the  rela- 
tive motions  in  longitude  and  latitude,  (k  and  n,}  during  some  short  interval  near 
the  middle  of  the  eclipse,  which  is  the  approximate  time  of  greatest  obscuration  j 
then  compute  the  inclination  of  the  relative  orbit  by  the  equation 

tang  I  =  |  ...  (122.)     (See  equa.  90) : 

after  which  ^  will  result  from  the  equation 

i/'  =  A  cos  I  ,  .  .  (123.)     (See  equa.  94). 

A  is  the  moon's  latitude  at  the  time  of  apparent  conjunction,  which  is  easily  tal. 
culated,  by  means  of  the  values  of  A:  and  n,  and  the  apparent  longitude  and  lati- 
tude of  the  moon,  found  for  some  instant  near  the  time  of  apparent  conjunction. 

For  the  beginning  and  end  of  the  total  eclipse,  we  have,  \p  =  appar.  semi-diam. 
of  moon —  appsir.  semi-diam.  of  sun  -f- 1"-5  ;  and  for  the  beginning  and  end  of  the 
annular  eclipse,  i^/=^  appar.  semi-diam.  of  sun  — appar.  semi-diam.  of  moon  —  1".5. 

485.  If  the  value  of  <//,  given  by  equation  (123,)  be  substituted  in  equation 
(121,)  tlu's  equation  will  make  known  the  time  of  greatest  obscuration;  but  this 
may  be  found  more  conveniently  by  a  different  process.     Let  NCF  (Fig.  82)  repre- 

.  82.  sent  a  portion  of  the  ecliptic,  EML  a  portion 

of  the  relative  orbit  passed  over  about  the 
time  of  greatest  obscuration,  C  the  stationa- 
ry position  of  the  sun's  centre,  and  M  the 
place  of  the  moon's  centre  at  the  instant  of 
its  nearest  approach  to  C.  Also,  let  a  =  CR 
the  apparent  distance  of  the  moon  from  the 
sun  in  longitude  at  the  time  of  the  nearest 
approach  of  the  centres,  A'  =  RM  the  moon's 
C  B.  N  apparent  latitude  at  the  same  time,  k  =  M& 

the  apparent  relative  motion  in  longitude  in  some  short  interval  about  this  time, 
and  n  =  krt  the  moon's  apparent  motion  in  latitude  during  the  same  interval.  The 
right-angled  triangles  Mnk  and  CMR  are  similar,  for  their  sides  are  respectively 
perpendicular  to  each  other ;  whence, 

M*  :  MR  :  :  kn  :  CR  ; 

and  CR  =  MR^L  or,  «  =  A'  |  .  .  .  (124). 

If  the  moon's  apparent  latitude  be  found  for  the  approximate  time  of  greatest 
obscuration,  and  substituted  for  A'  in  equation  (124,)  this  equation  will  give  very 
nearly  the  apparent  distance  (a)  of  the  two  bodies  in  longitude  at  the  true  time  of 
greatest  obscuration.  ^Vith  this,  and  the  apparent  distance  at  the  approximate 
time  of  greatest  obscuration,  together  with  the  relative  apparent  motion  in  longi- 
tude, the  true  time  of  greatest  obscuration  may  be  found  nearly  by  simple  propor- 
tion. A  more  accurate  result  may  then  be  had  by  finding  the  moon's  apparent 
latitude  for  the  time  obtained,  substituting  it  for  A'  in  equation  (124)  and  then  re- 
peating the  calculations. 

486.  A  simpler,  though  less  accurate  method  than  that  already  given,  of  find- 
ing the  times  of  beginning  and  end  of  the  total  or  annular  eclipse,  is  to  compute 
the  half  duration  of  the  total  or  annular  eclipse,  and  add  it  to,  and  subtract  it  from, 

*  Developing  the  radical  in  equation  (120,)  and  neglecting  all  the  terms  after 
tht  second,  as  being  very  small,  we  obtain  for  the  beginning  and  end  of  the  eclipse 
llie  mm*  convenient  formula 

_  1800s.  (A2  —  ^2) 

B 


OCCULTATIONS.  183 

the  time,  of  greatest  obscuration.  This  interval  may  easily  be  determined,  if  we 
can  find  the  rate  of  motion  on  the  relative  orbit,  and  the  distance  passed  over  by 
the  moon's  centre  during  the  interval.  Let  g,g'  (Fig.  82)  be  the  places  of  the 
moon's  centr«5  at  the  instants  of  the  two  interior  contacts,  and  M/i  the  distance 
passed  over  in  some  short  interval  (L).  Let  9  =  <  ~M.nk  the  complement  of  the 
inclination  of  the  relative  orbit,  k  =  MAr,  k'  =  Mn,  and  t  =  half  duration  of  total 
or  annular  eclipse.  The  triangles  MnA:,  CRM,  give 

JL-.  .  .  (125): 
sin  0 

•p  T»/r  \  l 

and  tang  RCM  =  tang  Mw&  =  -^,  or,  tang  0  =  -  ...  (126). 

Finding  the  value  of  0  by  the  last  equation,  and  substituting  it  in  equation  (125K 
we  obtain  the  value  of  k'  ;  and  then,  to  find  t,  we  have 


_A2     (Art.  484)  ; 


L  V  4,2  _  A2 

whence,  t  =  -      —  ^  -  =  -       —  -^  --  .  .  .  (127) 

487.  The  apparent  distance  of  the  centres  of  the  two  bodies  at  the  time  of  great- 
est obscuration  being  known,  the  quantity  of  the  eclipse  may  be  readily  found. 
We  have  but  to  subtract  the  apparent  distance  from  the  sum  of  the  apparent  semi- 
diameters,  and  state  the  proportion,  as  the  sun's  apparent  diameter  :  the  remain- 
der :  :  12  digits  :  the  digits  eclipsed.  (For  a  more  particular  description  of  the 
method  of  calculating  a  solar  eclipse,  see  Prob.  XXX.) 

OCCULTATIONS. 

488.  At  all  places  upon  the  earth's  surface,  which  at  a  given 
time  have  the  moon  in  the  horizon,  its  apparent  place  will  differ 
from  its  true  place,  by  the  amount  of  its  horizontal  parallax.     It 
follows,  therefore,  that  a  star  will  be  eclipsed  by  the  moon  some- 
where upon  the  earth,  in  case  its  true  distance  from  the  moon's 
centre  is  less  than  the  sum  of  the  moon's  semi-diameter  and  hori- 
zontal parallax. 

The  greatest  value  of  the  moon's  semi-diameter  is  1  6'  45",  and 
that  of  its  horizontal  parallax  61'  24".  If  we  add  the  sum  of  these 
numbers  to  5°  17'  34",  the  maximum  latitude  of  the  moon,  we  ob- 
tain as  the  result  6°  35'  43".  It  is  then  only  the  stars  which  have 
a  latitude  less  than  6°  35'  43"  that  can  experience  an  occn  Ration 
from  the  moon. 

489.  By  considering  the  various  situations  of  the  stars  liable  to  an  occultation, 
taking  the  greatest  and  least  values  of  the  sum  of  the  moon's  semi-diameter  and 
horizontal  parallax,  and  allowing  for  the  inequalities  of  the  motions  of  the  moon, 
it  has  been  found,  that,  if  at  the  time  of  the  mean  conjunction  of  the  moon  and 
a  star,  (that  is,  when  the  moon's  mean  longitude  is  the  same  with  the  longitude  of 
the  star,)  their  difference  of  latitude  exceed  1°  37',  there  cannot  be  an  occultation  ; 
if  the  difference  be  less  than  51',  there  must  be  an  occultation  somewhere  on  the 
earth  ;  and  that  between  these  limits  there  is  a  doubt,  which  can  only  be  removed 
by  the  calculation  of  the  mo&i's  true  place. 

490.  The  calculation  of  an  oecultation  is  very  nearly  the  same  as  that  of  a  solar 
eclipse.     The  only  difference  is  in  the  data.     The  star  has  no  diameter,  parallax, 
or  motion  in  longitude  ;  and  as  it  is  situated  without  the  ecliptic,  we  have,  in  place 
of  the  latitude  of  the  moon,  employed  in  solar  eclipses,   the  difference  between 


184         OF  THE  PLANETS  AND  THEIR  PHENOMENA. 

the  latitude  of  the  moon  and  that  of  the  star,  and  in  place  of  the  difference  be- 
tween the  longitudes  of  the  two  bodies  and  their  relative  hourly  motion  in  longi- 
tude, these  quantities  referred  to  an  arc  passing  through  the  star  and  parallel  to  the 
ecliptic.  Thus,  if  EC  (Fig.  81)  represent  the  ecliptic,  K  its  pole,  s  the  situation 
of  the  star,  M  that  of  the  moon,  and  sm1  an  arc  passing  through  s  and  parallel  to 
the  arc  EC,  we  have  in  place  of  mM,  m'M  =  mM  —  mm',  and  in  place  of  S-m,  s-m'. 
The  hourly  variation  of  Sm  must  also  be  reduced  to  the  arc  sm'. 

Fig.  83.  491-  The  reduction  of  the  difference  of  longitude  ef  the 

moon  and  star,  to  the  parallel  to  the  ecliptic,  p.issing 
through  the  star,  is  effected  by  multiplying  this  difference 
by  the  cosine  of  the  latitude  of  the  star.  For,  let  AB  (Fig. 
83)  be  an  arc  of  the  ecliptic,  and  A'B'  the  corresponding 
are  of  a  circle  parallel  to  it  ;  then,  since  similar  arcs  of  cir- 
_,/  cles  are  proportional  to  their  radii,  we  have 


BC  :  B'C'  :  :  AB  :  A'B' 


c  But,         B'C'  =  Co  =  B'C  cos  BCB'  =  BC  cos  BB' : 

.  AB.BCcosBB' 

hence,         A'B'  = -— =  AB  cos  BB'. 

BO 

The  reduction  of  the  relative  hourly  motion  in  longitude  to  the  parallel  in  ques- 
tion, is  obviously  effected  in  the  same  manner. 


CHAPTER   XVI. 

OF  THE  PLANETS,  AND  THE  PHENOMENA  OCCASIONED  BY  THEIR 
MOTIONS  IN  SPACE. 

APPARENT  MOTIONS  OF  THE  PLANETS  WITH  RESPECT  TO 

THE  SUN. 

492.  THE  apparent  motion  of  an  inferior  planet,  with  reference 
to  the  sun,  is  materially  different  from  that  of  a  superior  planet. 
The  inferior  planets  always  accompany  the  sun,  being  seen  alter- 
nately on  the  east  and  west  side  of  him,  and  never  receding  from 
him  beyond  a  certain  distance,  while  the  superior  planets  are  seen 
at  every  variety  of  angular  distance.  This  difference  of  apparent 
motion  arises  from  the  difference  of  situation  of  the  orbits  of  an 
inferior  and  superior  planet,  with  respect  to  the  orbit  of  the  earth  ; 
the  one  lying  within  and  the  other  without  the  earth's  orbit. 

Let  CAC'B  (Fig.  84)  represent  the  orbit  of  either  one  of  the  in- 
ferior planets,  Venus  for  example,  and  PKT  the  orbit  of  the  earth  ; 
which  we  will  suppose  to  be  circles,  and  to  lie  in  the  same  plane  ; 
and.  let  MLN  represent  the  sphere  of  the  heavens  to  which  all  bo 
dies  are  referred.     Suppose,  for  the  present,  that  the  earth  is  sta 
tionary  in  the  position  P,  and  through  P  draw  the  lines  PA,  PB, 
tangent  to  the  orbit  of  Venus,  and  prolong  them  on  till  they  inter 


APPARENT  MOTIONS  OF  THE  PLANETS. 


185 


sect  the  heavens  at  a  and  b.  When  Venus  is  at  C,  (the  earth  be- 
ing at  P,)  she  will  be  in  superior  conjunction,  and  when  at  C'  in 
inferior  conjunction.  Now,  by  inspecting  the  figure,  it  will  be  seen 
that  in  passing  from  C  to  C'  she  will  be  seen  in  the  heavens  on  the 
east  side  of  the  sun,  and  in  passing  from  C'  to  C  on  the  west  side 

Fig.  84. 


of  the  sun ;  also,  that  in  passing  from  C  to  A  she  will  recede  from 
the  sun  in  the  heavens,  from  A  to  C'  approach  him,  from  C'  to  B 
recede  from  him  again,  and  from  B  to  C  approach  him  again,  a 
and  6  will  be  her  positions  in  the  heavens  at  the  times  of  her  great- 
est eastern  and  western  elongations. 

When  Venus  is  to  the  east  of  the  sun,  she  is  seen  in  the  even- 
ing, and  called  the  Evening  Star ;  and  when  to  the  west,  she  is 
seen  in  the  morning,  and  called  the  Morning  Star. 

493.  We  have  in  the  foregoing  investigation  supposed  the  earth 
to  be  stationary,  a  supposition  which  is  contrary  to  the  fact ;  but 
it  is  plain  that  the  only  effect  of  the  earth's  motion  in  the  case  un- 
der consideration,  as  it  is  slower  than  that  of  the  planet,  is  to  cause 
the  points  A,  C',  B  to  advance  in  the  orbit,  without  altering  the 
nature  of  the  apparent  motion  of  the  planet  with  respect  to  the  sun. 
The  orbits  of  the  earth  and  planet  are  also  ellipses  of  small  eccen- 
tricity, and  are  slightly  inclined  to  each  other,  instead  of  being  cir- 
cles and  lying  in  the  same  plane  :  on  this  account,  as  the  greatest 
elongations  will  occur  in  various  parts  of  the  orbits,  they  will  differ 
in  value.    The  greatest  elongation  of  Venus  varies  from  45°  to  47° 
12'.     Its  mean  value  is  about  46°. 

494.  Owing  to  the  circumstance  of  the  orbit  of  Mercury  being 

24 


186         OF  THE  PLANETS  AND  THEIR  PHENOMENA. 

within  the  orbit  of  Venus,  the  greatest  elongation  of  this  planet  is 
less  than  that  of  Venus.  It  varies  between  the  limits  16°  12',  and 
28°  48';  and  is,  at  a  mean,  22°  30'. 

495.  Next,  suppose  PKT  (Fig.  84)  to  be  the  orbit  of  a  superior 
planet,  and  CAC'B  that  of  the  earth  ;  and,  as  the  velocity  of  the 
earth  is  much  greater  than  that  of  the  planet,  let  us,  for  the  present, 
regard  the  planet  as  stationary  in  the  position  P,  while  the  earth 
describes  the  circle  CAC'.     When  the  earth  is  at  C,  the  planet, 
being  at  P,  is  in  conjunction  with  the  sun.     When  the  earth  is  at 
A,  SAP,  the  elongation  of  the  planet  is  90°.     When  it  arrives  at 
C',  the  planet  is  in  opposition,  or  180°  distant  from  the  sun  :  and 
when  it  reaches  B,  the  elongation  is  again  90°.    At  intermediate 
points  the  elongation  will  have  intermediate  values.  If,  now,  we  re- 
store to  the  planet  its  orbitual  motion,  we  shall  manifestly  be  con- 
ducted to  the  same  results  relative  to  the  change  of  elongation,  as 
the  only  effect  of  such  motion  will  be  to  throw  the  points  A,  C',  B 
forward  in  the  orbit.     It  appears,  then,  that  in  the  course  of  a  sy- 
nodic revolution  a  superior  planet  will  be  seen  at  all  angular  dis- 
tances from  the  sun,  both  on  the  east  and  west  side  of  him.    From 
conjunction  to  opposition,  that  is,  while  the  earth  is  passing  from 
C  to  C',  the  planet  will  be  to  the  right,  or  to  the  west  of  the  sun  ; 
and  will  therefore  be  below  the  horizon  at  sunset,  and  rise  some 
time  in  the  course  of  the  night.     But,  from  opposition  to  conjunc- 
tion, or  while  the  earth  is  moving  from  C'  to  C,  it  will  be  to  the 
east  of  the  sun,  and  therefore  above  the  horizon  at  sunset. 

496.  To  find  the  length  of  the  synodic  revolution  of  a  planet.  — 
Let  us  first  take  an  inferior  planet,  Venus  for  instance.     Suppose 
we  assume,  at  a  given  instant,  the  sun,  Venus,  and  the  earth  to  be 
in  the  same  right  line  ;  then,  after  any  elapsed  time,  (a  day  for  in- 
stance,) Venus  will  have  described  an  angle  m,  and  the  earth  an 
angle  M  around  the  sun.     Now,  m  is  greater  than  M  ;  therefore 
at  the  end  of  a  day,  the  separation  of  Venus  from  the  earth,  (mea- 
suring the  separation  by  an  angle  formed  by  two  lines  drawn  from 
Venus  and  the  earth  to  the  sun,)  will  be  m  —  M  ;  at  the  end  of  two 
days  (the  mean  daily  motions  continuing  the  same)  the  angle  of 
separation   will  be   2  (m  —  M)  ;  at  the   end   of  three   days,    3 
(m  —  M)  ;  at  the  end  of  s  days,  s  (m  —  M).     When  the  angle  of 
separation  amounts  to  360°,  jhat  is,  when  s  (m  —  M)  =  360°,  the 
sun,  Venus,  and  the  earth  must  be  again  in  the  same  right  line, 
and  in  that  case 


. 

In  which  expression  s  denotes  the  mean  duration  of  a  synodic 
revolution,  m  and  M  being  taken  to  denote  the  mean  daily  motions. 

We  may  obtain  from  equation  (128)  another  equation,  in  which 
the  synodic  revolution  is  expressed  in  terms  of  the  sidereal  periods 
of  the  earth  and  planet. 


SYNODIC  REVOLUTIONS  OF  THE  PLANETS.          187 

Let  P  and  p  denote  the  sidereal  periods  in  question  ,  then,  since 

Id.  :  M°  :  :  P  :  360°, 
and  1      :.w    :  :  p  :  360  ; 

,,      360°  360°        ,     .      . 

M  =        -,  and  m  =  -  ;  substituting, 


**rj—f 

Equations  (128),  -(129),  although  investigated  for  an  inferio-j 
planet,  will  answer  equally  well  for  a  superior  planet,  provided  we 
regard  m  as  standing  for  the  mean  daily  motion  of  the  earth,  M  foi 
that  of  the  planet,  p  for  the  sidereal  period  of  the  earth,  and  P  foi 
that  of  the  planet.  For  the  earth  holds  towards  a  superior  planet. 
the  place  of  an  inferior  planet,  and  a  synodic  revolution  of  the  earth 
to  an  observer  on  the  planet,  will  obviously  be  a  synodic  revolu- 
tion of  the  planet  to  an  observer*  on  the  earth. 

497.  Equation  (128)  shows  that  the  length  of  a  mean  synodic 
revolution  depends  altogether  upon  the  amount  of  the  difference 
of  the  mean  daily  motions  of  the  earth  and  planet,  and  is  the  greater 
the  less  is  this  difference. 

It  follows  therefore  that  the  synodic  revolution  is  the  longest  for 
the  planets  nearest  the  earth. 

It  appears  by  equation  (129),  that  the  length  of  a  synodic  revo- 
lution is,  for  an  inferior  planet,  greater  than  the  sidereal  period  of 
the  planet,  and  for  a  superior  planet,  greater  than  the  sidereal  pe- 
riod of  the  earth.  The  actual  lengths  of  the  synodic  /evolutions 
of  the  different  planets  are  given  in  Table  V. 

498.  The  mean  synodic  revolution  of  a  planet  being  known,  and 
also  the  time  of  one  conjunction  or  opposition,  we  may  easily  as- 
certain its  mean  elongation  at  any  given  time,  and  thus  approxi- 
mately the  time  of  its  rising,  setting,  and  meridian  passage. 

499.  A  planet  will  rise  and  set  at  the  same  hours  at  the  end  of  a 
synodic  revolution  ;  and  will  be  an  evening  star,  that  is,  above  the 
horizon  at  sunset,  during  half  of  a  synodic  revolution,  and  a  morn- 
ing star,  that  is,  above  the  horizon  at  sunrise,  during  an  equal  in- 
terval of  time.     The  inferior  planets  will  be  evening  stars  from 
superior  to  inferior  conjunction  ;  and  the  superior  planets  from  op- 
position to  conjunction. 

Mercury  is  an  evening  star  for  a  period  of  2  months  ;  Venus 
during  an  interval  of  9|  months  ;  Mars  for  1  year  and  1  month  ; 
Jupiter  for  6^  months  ;  Saturn  and  Uranus  each  a  few  days  more 
than  6  months. 

STATIONS  AND  RETROGRADATIONS  OF  THE  PIANETS. 

500.  The  apparent  motions  of  the  planets  in  the  heavens,  as  has 
already  been  stated  (13),  are  not,  like  those  of  the  sun  and  moon, 


188 


OF  THE  PLANETS  AND  THEIR  PHENOMENA. 


continually  from  west  to  east,  or  direct,  but  are  sometimes  also 
from  east  to  west,  or  retrograde.  The  retrograde  motion  takes 
place  over  arcs  of  but  a  small  number  of  degrees ;  and  in  changing 
the  direction  of  their  motions,  the  planets  are  for  several  days  sta- 
tionary in  the  heavens.  These  phenomena  are  called  the  Stations 
and  Retrogradations  of  the  planets.  We  now  propose  to  inquire 
theoretically  into  the  particulars  of  the  motions  in  question,  and  to 
show  how  the  phenomena  just  mentioned  result  from  the  motions 
of  the  planets  in  connection  with  the  motion  of  the  earth. 

Let  CAC'B  (Fig.  84,  p.  185)  represent  the  orbit  of  an  inferior 
planet,  and  PKT  the  orbit  of  the  earth ;  both  considered  as  circles, 
and  as  situated  in  the  same  plane.  If  the  earth  were  continually 
stationary  in  some  point  P  of  its  orbit,  it  is  plain  that  while  the 
planet  was  moving  from  B  the  position  of  greatest  western  elonga- 
tion to  A  the  position  of  greatest  eastern  elongation,  it  would  ad- 
vance in  the  heavens  from  6  to  a ;  that,  while  it  was  moving  from 
A  to  B,  that  is,  from  greatest  eastern  to  greatest  western  elonga- 
tion, it  would  retrograde  in  the  heavens  from  a  to  b ;  and  that,  in 
passing  the  points  A  and  B,  as  it  would  be  moving  directly  towards 
or  from  the  earth,  it  would  for  a  time  appear  stationary  in  the 
heavens  in  the  positions  a  and  b. 

But  the  earth  is  in  fact  in  motion,  and  the  actual  apparent  mo- 
tion of  the  planet  is  in  consequence  materially  different  from  this. 
Let  A,  A7  (Fig.  85)  be  the  positions  of  the  planet  and  earth  at  the 
time  of  the  greatest  eastern  elongation,  C',  P  their  positions  at  in- 

.  85. 


ferior  conjunction,  and  B,  B7  their  positions  at  the  greatest  western 
elongation.     At  the  time  of  the  greatest  eastern  elongation,  while 


STATIONS  AND  RETROGRADAT10NS  OF  THE  PLANETS.    189 

the  planet  describes  a  certain  distance  AD  on  the  line  of  the  cen- 
tres of  the  earth  and  planet,  the  earth  moves  forward  in  its  orbit  a 
certain  distance  A'D' ;  so  that,  instead  of  appearing  stationary  at  a 
in  the  interval,  the  planet  will  advance  in  the  heavens  from  a  to  d. 
From  the  same  cause  it  will  have  a  direct  motion  about  the  time 
of  the  greatest  western  elongation.  As  it  advances  from  A  towards 
C',  the  direct  motion  will  continue  ;  but,  as  the  daily  arc  described 
by  the  planet  will  make  a  less  and  less  angle  with  the  daily  arc  de- 
scribed by  the  earth,  the  rate  of  motion  will  continually  decrease, 
and  finally,  when  the  planet  has  come  into  a  position  with  respect 
to  the  earth,  such  that  the  lines  of  direction  of  the  planet,  mp,  m'pf, 
at  the  beginning  and  end  of  the  day  are  parallel,  it  will  be  station 
ary  in  the  heavens.  As  the  daily  arc  of  the  planet  is  greater  than 
that  of  the  earth,  and  becomes  parallel  to  it  in  inferior  conjunction, 
the  planet  will  be  in  the  position  in  question  before  it  comes  into 
inferior  conjunction. 

Subsequent  to  this,  the  inclination  of  the  daily  arcs  still  dimin- 
ishing, the  lines  of  direction  of  the  planet  at  the  beginning  and  end 
of  the  day  will  diverge,  and  therefore  the  motion  will  be  retro- 
grade. After  inferior  conjunction,  the  inclination  of  the  arcs  will, 
at  corresponding  positions  of  the  earth  and  planet,  obviously  be  the 
same  as  before.  It  follows,  therefore,  that  the  planet  will  be  at  its 
western  station  when  it  is  at  the  "same  angular  distance  from  the 
sun  as  at  its  eastern  station  ;  that  its  motion  will  be  retrograde  un- 
til it  has  passed  inferior  conjunction  and  arrived  at  its  western  sta- 
tion ;  and  that  after  this  it  will  be  direct,  q  and  n  represent  the 
positions  of  the  planet  and  the  earth  at  the  time  of  the  western  sta- 
tion ;  C'q  =  C'p,  and  Pn  =  Pm. 

The  diminution  of  the  elongation  of  the  planet  at  its  two  stations 
is  not  the  only  effect  of  the  earth's  motion  in  the  case  under  con- 
sideration ;  it  also  accelerates  the  direct,  and  retards  the  retrograde 
motion  of  the  planet,  and  gives  to  the  planet  along  with  the  sun  an 
apparent  motion  of  revolution  around  the  earth. 

501.  Let  us  now  pass  to  the  case  of  a  superior  planet. .  Sup- 
pose AC'B  (Fig.  85)  to  be  the  orbit  of  the  earth,  and  A'PB'  that 
of  the  planet.  Since  the  earth  is  an  inferior  planet  to  an  observer 
stationed  upon  a  superior  planet,  it  appears  by  the  foregoing  arti- 
cle that  it  will,  to  an  observer  so  situated,  have  a  retrograde  mo- 
tion while  it  is  passing  over  a  certain  arc  pC'q  in  the  inferior  part 
of  its  orbit,  and  a  direct  motion  during  the  remainder  of  the  sy- 
nodic revolution.  Now,  it  is  plain  that  the  direction  of  the  planet's 
motion,  as  seen  from  the  earth,  will  always  be  the  same  as  the  di- 
rection of  the  earth's  motion  as  seen  from  the  planet.  When  the 
earth  is  at  C',  the  middle  of  the  arcpC'^,  the  planet  is  in  opposi- 
tion. It  follows,  therefore,  that  a  superior  planet  has  a  retrograde 
motion  during  a  small  portion  of  its  .synodic  revolution,  about  the 
time  of  opposition.  (See  Table  V.) 


190 


OF  THE  PLANETS  AND  THEIR  PHENOMENA. 


O 


PHASES  OF  THE  INFERIOR  PLANETS. 

502.  To  the  naked  sight  the  disc  of  the  planet  Venus  appears 
circular,  like  that  of  each  of  the  other  planets,  but  the  telescope 
shows  this  to  be  an  optical  illusion.     When  Venus  is  repeatedly 
observed  with  a  telescope,  it  is  seen  to  present  in  its  various  posi- 
tions with  respect  to  the  sun  the  same  variety  of  phases  as  the 
moon ;  being  a  full  circle  at  superior  conjunction,  a  half  circle  at 
the  greatest  eastern  and  western  elongations,  and  a  crescent,  with 
the  horns  turned  from  the  sun,  before  and  after  inferior  conjunction. 

Mercury  exhibits  precisely  similar  phases,  but  being  smaller,  at 
a  greater  distance  from  the  earth,  and  much  nearer  the  sun,  its 
phases  are  not  so  easily  observed  as  those  of  Venus. 

503.  The  phases  of  Venus  are  easily  accounted  for,  by  suppos- 
ing it  to  be  an  opake  spherical  body,  and  to  shine  by  reflecting  the 
sun's  light,  and  by  taking  into  consideration  its  motion  with  respect 

to  the  sun  and  earth.  The  hemi- 
sphere turned  towards  the  sun  is 
illuminated  by  him,  and  the  other  is 
in  the  dark,  and  as  the  planet  re- 
volves around  the  sun,  various  por- 
tioiis  of  the  enlightened  half  are 
turned  towards  the  earth  :  in  supe- 
rior conjunction,  the  whole  of  it ;  at 
the  greatest  elongations,  one  half; 
and  near  inferior  conjunction,  but  a 
small  part.  This  will  be  abundant- 
ly evident  on  inspecting  Fig.  86. 
The  phases  corresponding  to  the 
positions  represented  are  delineated 
in  the  figure. 

The  phases  of  Mercury  are  ob- 
viously susceptible  of  a  similar  ex- 
planation. 

504.  The  disc  of  the  planet  Mars  also  undergoes  changes  of 
form,  but  they  are  of  comparatively  moderate  extent.     It  is  some- 
times gibbous,  but  never  has  the  form  of  a  crescent.     Indeed,  on 
the  .supposition  that  Mars  is  an  opake  body  illuminated  by  the  sun, 
we  would  not  see  the  whole  of  the  enlightened  hemisphere,  except 
in  conjunction  and  opposition,  but  there  would  always  be  more 
tlinn  half  of  it  turned  towards  the  earth,  and  therefore  the  disc 
bhould  always  be  larger  than  a  half  circle. 

505.  The  discs  of  the  other  superior  planets  do  not  experience 
any  perceptible  variation  of  form,  for  the  reason,  doubtless,  that 
their  orbits  are  so  large  with  respect  to  the  orbit  of  the  earth,  that 
all,  or  very  nearly  all  of  their  illuminated  hemispheres,  is  con- 
stantly visible  from  the  earth. 


TRANSITS  OF  THE  INFERIOR  PLANETS.  191 


TRANSITS  OF  THE  INFERIOR  PLANETS. 

506  The  two  inferior  planets  Venus  and  Mercury,  at  inferior 
conjunction,  sometimes,  though  rarely,  pass  between  the  sun  and 
earth,  and  are  seen  as  a  dark  spot  crossing  the  sun's  disc.  This 
phenomenon  is  called  a  Transit.  It  will  take  place,  in  the  case 
of  either  planet,  whenever,  at  the  time  of  inferior  conjunction,  it  is 
so  near  either  node  that  its  geocentric  latitude  is  less  than  the  ap- 
parent semi-diameter  of  the  sun. 

507.  The  transits  of  Venus  take  place  alternately  at  intervals  of 
8  and  1051  Or  121|  years.     The  last  were  in  the  years  1761  and 
1769.     The  next  will  be  in  1874  and  1882 ;  of  which  the  latter 
will  be  visible  in  this  country. 

In  consequence  of  the  greater  distance  of  Mercury  from  the 
earth,  a  greater  portion  of  its  orbit  is  directly  interposed  between 
the  sun  and  earth  than  of  the  orbit  of  Venus  ;  moreover,  the  sy- 
nodic revolution  of  Mercury  is  shorter  than  that  of  Venus.  On 
these  accounts,  it  happens  that  the  transits  of  Mercury  are  much 
more  frequent  than  those  of  Venus.  The  last  transit  of  Mercury 
was  on  May  8th,  1845.  The  next  two  will  take  place  in  1848, 
and  1861,  in  the  month  of  November.  The  first,  which  will  oc- 
cur on  the  9th,  will  be  visible  in  this  country. 

508.  A  transit  is  calculated  in  a  precisely  similar  manner  with'  a 
solar  eclipse  ;  the  planet  in  the  one  calculation  answering  to  the 
moon  in  the  other. 

509.  A  transit  is  an  important  phenomenon  in  a  practical  point 
of  view,  as  it  furnishes  the  most  exact  means  we  possess  of  ascer- 
taining the  sun's  parallax.     In  order  to  understand  how  this  phe- 
nomenon can  be  used  for  this  purpose,  we  have  only  to  consider 
that,  in  consequence  of  the  difference  of  the  parallaxes  of  the  sun 
and  Venus,  observers  at  different  stations  upon  the  earth  will  refer 
the  planet  to  different  points  upon  the  sun's  disc,  and  that  therefore, 
to  such  observers,  the  transit  will  take  place  along  different  chords, 
and  be  accomplished  in  unequal  portions  of  time.    This  fact  is  rep- 
resented to  the  eye  in  Fig.  87.  It  is  then  to  be  expected,  that,  if  the 
durations  of  the  transit  at  two  different  places  should  be  noted,  the 

Fig.  87. 


difference  of  the  parallaxes   of  the  sun  and  Venus,  upon  which 
alone  the  difference  of  the  duration  depends,  could  be  computed. 
This  computation  is  in  fact  possible.    Also,  the  ratio  of  the  paral 
sixes  being  inversely  as  that  of  the  distances,  could  be  found  by  the 


192        OF  THE  PLANETS  AND  THEIR  PHENOMENA. 

elliptical  theory  of -the  planetary  motions,  and  thus  the  parallax  both 
of  the  sun  and  Venus  would  become  known. 

510.  The  parallax  of  the  sun,  as  it  is  now  known,  was  deduced 
from  observations  upon  the  transits  of  Venus  in  1769  and  1761. 
Expeditions  were  fitted  out  on  the  most  efficient  scale,  by  the 
British,  French,  Russian,  and  other  governments,  and  sent  to  va- 
rious parts  of  the  earth,  remote  from  each  other,  to  observe  the 
transit  of  1769,  that  the  parallax  of  the  sun  might  be  computed 
from  the  results  of  the  observations.     The  sun's  parallax,  as  de- 
termined by  Professor  Encke  from  the  observations  made  upon  the 
transit  in  question,  and  that  of  1761,  is  8".5776.  ft" it 2. 

APPEARANCES,   DIMENSIONS,  ROTATION,  AND  PHYSICAL 
CONSTITUTION  OF  THE  PLANETS. 

511.  It  appears  from  admeasurement  with  the  telescope  and 
micrometer,  that  the  apparent  diameter  of  a  planet  is  subject  to 
sensible  variations.     The  apparent  diameter  of  Venus,  as  well  as 
of  Mercury,  is  greatest  in  inferior  conjunction,  and  least  in  superior 
conjunction  ;  while  the  apparent  diameter  of  each  of  the  other 
planets  is  greatest  in  opposition  and  least  in  conjunction.     These 
variations  of  the  apparent  diameters  of  the  planets,  are  necessary 
consequences  of  the  changes  that  take  place  in  the  distances  of 
the  planets  from  the  earth.  (See  Fig.  84.) 

51 2.  The  real  diameter  of  a  planet  is  deduced  from  its  apparent 
diameter  and  horizontal  parallax.     (See  Art.  429.)     When  the  di- 
ameters of  the  planets  have  been  found,  their  relative  surfaces  and 
volumes  are  easily  obtained ;  for  the  surfaces  are  as  the  squares 
of  the  diameters,  and  the  volumes  as  the  cubes. 

513.  The  order  of  magnitude  of  the  planets  is  as  follows: 
1  Jupiter,  2  Saturn,  3  Uranus,  4  the  Earth,  5  Venus,  6  Mars, 
7  Mercury,  8  Pallas,  9  Ceres,  10  Juno,  11  Vesta.     The  range  of 
magnitude,  for  the  principal  planets,  is  from  1  to  about  20,000. 
(The  relative  magnitudes  of  the  planets  are  represented  to  the  eye 
in  the  Frontispiece.)     (See  Note  VIII.) 

514.  Spots  more  or  less  dark  have  been  seen  upon  the  discs  of 
most  of  the  principal  planets  ;  and  by  passing  across  them  from 
east  to  west  and  reappearing  at  the  eastern  limbs,  have  established 
that  the  planets  upon  which  they  are  observed  rotate  upon  axes 
from  west  to>east.     From  repeated  careful  observations  upon  the 
situations  of  these  spots,  the  periods  of  rotation,  and  the  positions 
of  the  axes,  have  been  determined.     (See  Note  IX.) 

The  periods  of  rotation  of  Mercury,  Venus,  the  Earth,  and  Mars, 
are  all  about  24  hours,  and  of  Jupiter  and  Saturn  about  10  hours. 
Those  of  the  other  planets  are  not  known.  The  axes  of  rotation 
remain  continually  parallel  to  themselves,  as  the  planets  revolve  in 
their  orbits. 

515.  The  amount  of  light  and  heat,  which  the  sun  bestows  upon 


MERCURY VENUS.  193 

the  planets,  decreases  as  we  recede  from  the  sun,  in  the  same  ratio 
that  the  square  of  the  distance  increases.     (See  Table  IV.) 

516  It  will  be  seen  in  the  sequel  that  the  planets  are  all  opake 
bodies,  like  the  earth  ;  and  that  they  are  surrounded  with  an  atmo- 
sphere, after  the  same  manner  as  the  earth. 

MERCURY. 

517.  In  consequence  of  its  proximity  to  the  sun,  Mercury  is 
rarely  visible  to  the  naked  eye.  When  seen  under  the  most  favor- 
able circumstances  about  the  time  of  greatest  elongation,  it  presents 
the  appearance  of  a  star  of  the  3d  or  4th  magnitude.  Its  phases 
show  that  it  is  opake,  and  illuminated  by  the  sun.  Its  apparent 
diameter  varies  with  its  distance  from  5"  to  12".  Its  real  diame- 
ter is  about  3000  miles,  or  f  of  that  of  the  earth,  and  its  volume 
is  about  TV  of  the  earth's  volume.* 

Mercury  performs  a  rotation  on  its  axis  in  24h.  S^m.,  and  its 
axis  is  inclined  to  the  ecliptic  under  a  small  angle. 

518.  Owing  to  the  dazzling  splendor  of  its  rays,  and  the  tremulous  motion  in- 
duced by  the  ever-varying  density  of  the  air  and  vapors  near  the  earth's  surface, 
through  which  it  is  seen,  the  telescope  does  not  present  a  well-defined  image  of  the 
disc  of  this  planet.  Schroeter  is  the  only  observer  who  has  ever  detected  any  spots 
upon  it.  From  the  fact  that  spots  are  only  occasionally  seen,  it  has  been  inferred 
that  the  planet  is  surrounded  with  a  dense  atmosphere,  which  reflects  a  strong  light, 
and,  except  when  it  is  particularly  pure,  prevents  the  darker  body  of  the  planet 
from  being  seen. 

Schroeter,  in  making  observations  upon  Mercury  at  the  time  his  disc  had  the 
form  of  a  crescent,  discovered  that  one  of  the  horns  of  the  crescent  became  blunt 
at  the  end  of  every  24  hours  :  from  which  he  inferred  that  the  planet  turned  upon 
an  axis,  and  had  mountains  upon  its  surface,  which  were  brought  at  the  end  of 
every  rotation  into  the  same  position  with  respect  to  his  eye  and  the  sun. 

VENUS. 

519.  Venus  is  the  most  brilliant  of  all  the  planets,  and  generally 
appears  larger  and  brighter  than  any  of  the  fixed  stars.     At  times, 
it  emits  so  much  light  as  to  be  visible  at  noonday.     It  is  found  by 
calculation,  that  the  epochs  in  the  course  of  a  synodic  revolution, 
at  which  Venus  gives  most  light  to  the  earth,  are  those  at  which, 
being  in  the  inferior  part  of  its  orbit,  it  has  an  elongation  of  about 
40°.     They  are  about  36  days  before  and  after  inferior  conjunc- 
tion.    The  disc  is  then  considerably  less  than  a  semicircle,  but  the 
increased  proximity  to  the  earth  more  than  compensates  for  the 
diminished  size  of  the  disc.     Venus  will  besides  attain  to  greater 
splendor  in  some  revolutions  than  others,  in  consequence  of  being 
nearer  the  earth,  when  in  the  most  favorable  position. 

520.  As  seen  through  a  telescope,  Venus  presents  a  disc  of 
nearly  uniform  brightness,  and  spots  have  very  rarely  been  seen 
upon  it.     Its  phases  prove  it  to  be   an  opake  spherical  body, 
shining  by  reflecting  the  sun's  light.     Its  apparent  diameter  varies 
with  its  distance  from  10"  to  61".     Its  real  diameter  is  about  7800 

*  The  exact  diameters,  volumes,  times  of  rotation,  &C.,.  of  the  different  planets*, 
as  far  as  known,  may  be  found  in  Table  IV. 

25 


194  OF  THE  PLANETS  AND  THEIR  PHENOMENA. 

miles,  and  its  volume  about  ^y less  than  that  of  the  earth.  The 
period  of  its  rotation  is  23h.  21m.  The  inclination  of  its  axis  tc 
the  plane  of  its  orbit  is  not  exactly  known,  but  is  not  far  from  18°. 

521.  From  the  remarkable  vivacity  of  the  light  of  this  planet,  which  far  ex- 
ceeds  that  of  the  light  reflected  from  the  moon's  surface,  as  well  as  the  transitory 
nature  of  the  few  darkish  spots  which  have  been  seen  upon  its  disc,  it  is  inferred 
that  it  is  surrounded  by  a  dense  and  highly  reflective  atmosphere,  which  in  gene- 
ral screens  the  whole  of  the  darker  body  of  the  planet  from  our  view.     The  truth 

88.  of  this   inference  is  confirmed  by  certain  deli- 

cate observations  made  by  Schroeter.  This 
astronomer  distinctly  discerned  a  faint  bluish 
light  stretching  beyond  the  proper  termination 
of  one  of  the  horns  of  the  crescent  into  the  dark 
part  of  the  face  of  the  planet,  as  is  represented 
in  Fig.  88,  where  the  left  extremity  of  the  dot- 
ted line  represents  the  natural  terminating  point 
of  one  of  the  horns  of  the  crescent.  This  he 
considered  to  be  a  twilight  on  the  surface  of 
Venus. 

Since  the  transparency  of  Venus's  atmosphere 
is  variable,  becoming  occasionally  such  as  to 
admit  of  the  body  of  the  planet's  being  seen 
through  it,  we  must  suppose  that  it  contains 
aqueous  vapor  and  clouds,  and  therefore  that 
there  are  bodies  of  water  upon  the  surface  of  the  planet.  It  is  in  fact  supposed 
that  isolated  clouds  have  actually  been  seen.  The  most  natural  explanation  of 
the  bright  spots  which  have  sometimes  been  noticed  on  the  disc  is,  that  they  are 
clouds  more  highly  reflective  than  the  atmosphere  or  than  the  clouds  in  general. 

522.  There  are  great  inequalities  on  the  surface  of  Venus,  and,  it  would  seem, 
mountains  much  higher  than  any  upon  our  globe.  Schroeter  detected  these  masses 
by  several  infallible  marks.     In  the  first  place  the  edge  of  the  enlightened  part  of 
Venus  is  shaded,  as  seen  in  Figs.  88,  89,  and  90,  and  as  the  moon  appears  when 
in  crescent  even  to  the  naked  eye.     This  appearance  is  doubtless  caused  by  shad- 
ows cast  by  mountains  ;  which  are  naturally  best  seen  on  that  part  of  the  planet 
to  which  the  sun  is  rising  or  setting,  where  they  are  longest.     In  the  next  place, 
the  edge  of  the  disc  shows  marked  irregularities.     Thus  it  often  appears  rounded 
at  the  corners,  as  in  Fig.  89,  owing  undoubtedly  to  part  of  the  disc  being  rendered 
invisible  there  by  the  shadow  or  interposition  of  some  line  of  eminences  ;  and  at 

Fig.  89.  Fig.  90. 


. 

other  times,  as  in  Fig.  90,  a  single  bright  point  appears  detached  from  the  disc— 
the  top  of  a  high  mountain,  illuminated  across  a  dark  valley. 

Schroeter  found  that  theke  appearances  recurred  regularly  at  equal  intervals  of 
about  23$  hours  ;  the  same  period  as  that  which  Cassini  had  previously  found  for 
the  completion  of  a  rotation,  by  observations  upon  the  spots. 


• 


MARS.  195 

MARS. 

523  Mars  is  of  the.  apparent  size  of  a  star  of  the  first  or  second 
magnitude,  and  is  distinguished  from  the  other  planets  by  its  red 
and  fiery  appearance.  Ihe  observed  variation  in' the  form  of  its 
disc  (504)  shows  that  it  derives  its  light  from  the  sun.  Its'  greatest 
and  least  apparent  diameters  are  respectively  4"  and  18".  Its  real 
diameter  is  something  over  4000  miles,  or  rather  more  than  |  of 
the  diameter  of  the  earth,  and  its  bulk  is  about  |  of  that  of  the 
earth.  ^ 

:  Mars  revoIVes'on'its  a'xis'in  24h.  37m. ;  and  its'  axis  is  incliAed 
to  the  ecliptic  in  an  angle  of  about  60°.  It  appears,  from  meas- 
urements made  with  the  micrometer,  that  its  polar  diameter  is  less 
than  the  equatorial,  and  thus,  that,  like  the  earth,  it  is  flattened  at 
its  poles.  According  to  Sir  W.  Herschel,  its  oblateness  (159) 
is.  TV '  according  to  Arago  ^T. 

524.  When  the  disc  of  Mars  is  examined  with  telescopes  of 
great  power  it  is  generally  seen  to  be  diversified  with  spots  of  dif- 
ferent shades,  which,  with  occasional  variations,  retain  constantly 
the  same  size  and  form. 

They  are  conjectured  to  be  continents  and  seas.  In  fact,  Sir  J.  F.  W.  Her- 
schel has  on  several  occasions,  in  examining  this  planet  with  a  good  telescope,  no- 
ticed that  some  of  its  spots  are  of  a  reddish  color,  while  others  have  a  greenish 
tinge.  The  former  he  supposes  to  be  land,  and  the  latter  water.  Fig.  91  repre- 
sents Mars  in  its  gibbous  state  as  Yig.  91. 
seen  by  Herschel  in  his  20  feet  re- 
flector, on  the  16th  of  August, 
1'830.  The  darker  parts  are  seas. 
The  bright  spot  at  the  top  is  at 
one  of  the  poles  of  Mars.  At 
other  times  a  similar  bright  spot  is 
seen  at  the  other  pole.  These 
brilliant  white  spots  have  f  been 
conje6tured  with  a  great  deal  of 
probability  to  be  snow  ;  as  they 
are  reduced  in  size,  and  sometimes 
disappear  when  they  have  been 
long  exposed  to  the  sun,  and  are 
greatest  when  just  emerging  from 
the  long  night  of  their  polar  winter. 

525.  The  great  divisions  of  the 
surface  of  Mars  are  seen  with  dif- 
ferent  degrees  of  distinctness    at 

different  times,  arid  sometimes  disappear,  either  partially  or  entirely  :  parts  oi  tun 
disc  also  appear  at  times  particularly  dark  or  bright.  From  these  facts  it  is  to 'be 
inferred  that  this  planet  is  environed  with  an  atmosphere,  and  that  this  contains 
aqueous  vapor  which,  by  varying  in  quantity  and  density,  renders  its  transpa- 
rency variable.  ;  ^.  ... 

526.  No  mountains  have  been  detected  upon  Mars.     But  this  is  no  good  reason 
for  supposing  that  they  a,re  really  wanting  there  ;  for,  if  the  surface  of  Mars  be 
actually  diversified  with  mountains  and  valleys,  since  its  disc  never  differs  much 
from  a  full  circle,  we  have  no  reason  to  expect  that  its  edge  would  present  that 
shaded  appearance  and  those  irregularities  which  have  been  noticed  on  Venus  and 
Mercury,  when  of  the  form  of  a  crescent.     The  same  remarks  will  apply  with  still 
greater  force  to  the  other  superior  planets. 

527.  The  ruddy  color  of  the  light  of  Mars  has  generally  been  attributed  to  its 


196         OF  THE  PLANETS  AND  THEIR  PHENOMENA. 

atmosphere,  but  Sir  John  Herschel  finds  a  sufficient  cause  for  this  phenomenon  in 
the  ochrey  tinge  of  the  general  soil  of  the  planet  (524.) 

JUPITER  AND  ITS  SATELLITES. 

528.  Jupiter  is  the  most  brilliant  of  the  planets,  except  Venus, 
and  sometimes  even  surpasses  Venus  in  brightness.     The  eclipses 
of  its  satellites  prove  that  it  is  an  opake  body,  and  that  it  shines 
by  reflecting  the  light  of  the  sun.     Its  apparent  diameter,  when 
greatest,  is  46",  and  when  least,  30". 

Jupiter  is  the  largest  of  all  the  planets.  Its  diameter  is  about 
1 1  times  the  diameter  of  the  earth,  or  about  87,000  miles,  and  its 
bulk  is  more  than  1200  times  that  of  the  earth.  It  turns  on  an 
axis  nearly  perpendicular  to  the  ecliptic,  and  completes  a  rotation 
in  9h.  56m.  The  polar  diameter  is  about  T\  less  than  the  equa- 
torial. 

529.  When  Jupiter  is  examined  with  a  good  telescope,  its  disc 
is  always  observed  to  be  crossed  by  several  obscure  spaces,  which 
are  nearly  parallel  to  each  other,  and  to  the  plane  of  the  equator. 

.  90  .  These  are  called  the  Belts  of 

Jupiter.  (See  Fig.  92,  which 
represents  the  appearance  of 
Jupiter  as  seen  by  Sir  John 
Herschel  in  his  twenty-feet 
reflector,  on  the  23d  of  Sep- 
tember, 1832.)  They  vary 
somewhat  in  number,  breadth, 
and  situation  on  the  disc,  but 
never  in  direction.  Sometimes 
only  one  or  two  are  visible ;  on 
other  occasions  as  many  as 
eigh^  have  been  seen  at  the 
same  time.  Sir  William  Her- 
schel even  saw  them  on  one  or 
two  occasions  broken  up  and  distributed  over  the  whole  face  of  the 
planet :  but  this  phenomenon  is  extremely  rare.  Branches  run- 
ning out  from  the  belts  and  subdivisions,  as  represented  in  the 
figure,  are  by  no  means  uncommon.  Dark  spots  of  invariable 
form  and  size  have  also  been  seen  upon  them.  These  have  been 
observed  to  have  a  rapid  motion  across  the  disc,  and  to  return  at 
equal  intervals  to  the  same  position  on  the  disc,  after  the  same 
manner  as  the  sun's  spots;  which  leaves  no  room  to  doubt  that  they 
are  on  the  body  of  the  planet,  and  that  this  turns  upon  an  axis. 
Bright  spots  have  also  been  noticed  upon  the  belts.  The  belts 
generally  retain  pretty  nearly  the  same  appearance  for  several 
months  together,  but  occasionally  marked  changes  of  form  and 
size  have  taken  place  in  the  course  of  an  hour  or  two. 

The  occasional  variations  of  Jupiter's  belts,  and  the  occurrence  of  spots  upon 
them,  which  are  undoubtedly  permanent  portions  of  the  mass  of  the  planet,  render 
it  extremely  probable  that  thpy  are  the  body  of  the  planet  seen  through  an  atmo. 


JUPITER — SATURN.  197 

sphere  of  variable  transparency  ;  but  in  general  having  extensive  tracts  of  compar- 
atively clear  sky  in  a  direction  parallel  to  the  equator.  These  are  supposed  to  b« 
determined  by  currents  analogous  to  our  trade  winds,  but  of  a  much  more  steady 
and  decided  character;  as  would  be  the  necessary  consequence  of  the  superior 
velocity  of  rotation  of  this  planet.  As  remarked  by  Herschel,  that  it  is  the  com- 
paratively darker  body  of  the  planet  which  appears  in  the  belts,  is  evident  from 
this, — that  they  do  not  come  up  in  all  their  strength  to  the  edge  of  the  disc,  but 
fade  away  gradually  before  they  reach  it. 

The  bright  belts,  intermediate  between  the  dark  ones,  are  probably  bands  of 
clouds  or  tracts  of  less  pure  air. 

530.  The  satellites  of  Jupiter,  as  it  has  been  already  remarked, 
are  visible  with  telescopes  of  very  moderate  power.     With  the 
exception  of  the  second,  which  is  a  little  smaller,  they  are  some- 
what larger  than  the  moon.     The  orbits  of  the  satellites  lie  very 
nearly  in  the  plane  of  Jupiter's  equator.     They  are  therefore  all 
viewed  nearly  edgewise  from  the  earth,  and  in  consequence  the 
satellites  always  appear  nearly  in  a  line  with  each  other. 

531.  Sir  W.  Herschel,  in  examining  the  satellites  of  Jupiter 
with  a  telescope,  perceived  that  they  underwent  periodical  varia- 
tions of  brightness.    These  variations  he  supposed  to  proceed  from 
a  rotation  of  the  satellites  upon  axes,  which  caused  them  to  turn 
different  faces  towards  the  earth  ;  and  from  repeated  and  careful 
observations  made  upon  them,  he  discovered  that  each  satellite 
made  one  turn  upon  its  axis  in  the  same  time  that  it  accomplished 
a  revolution  around  the  primary ;  and  therefore,  like  the  moon, 
presented  continually  the  same  face  to  the  primary. 

SATURN,  WITH  ITS    SATELLITES  AND  RING. 

532.  Saturn  shines  with  a  pale  dull  light.     Its  apparent  diame- 
ter varies  only  3"  or  4"  by  reason  of  the  change  of  distance,  and 
is  at  the  mean  distance  about  16".     The  eclipses  of  its  satellites 
prove  that  it  is  opake  and  illuminated  by  the  sun. 

Saturn  is  the  largest  of  the  planets,  next  to  Jupiter.  Its  diame- 
ter is  about  10  times  the  diameter  of  the  earth,  or  79,000  miles; 
and  its  volume  is  about  900  times  that  of  the  earth.  The  rotation 
on  its  axis  is  performed  in  lOh.  29m.  The  inclination  of  its  axis 
to  the  ecliptic  is  about  60°.  Its  oblateness  is  TV 

533.  The  disc  of  Saturn,  like  that  of  Jupiter,  is  frequently 
crossed  with  dark  bands  or  belts,  in  a  direction  parallel  to  its  equa- 
tor.    Extensive  dusky  spots  are  also  occasionally  seen  upon  its 
surface.  (See  Fig.  93.) 

The  cause  of  Saturn's  beltsus  doubtless  the  same  as  of  Jupiter's.  They  accord- 
ingly prove  the  existence  of  an  atmosphere  and  of  aqueous  vapor,  and  thus  also  of 
bodies  of  water,  upon  the  surface  of  Saturn. 

534.  The  planet  Saturn   is   distinguished  from  all  the  other 
planets  in  being  surrounded  by  a  broad,  thin,  luminous  ring,  situ- 
ated in  the  plane  of  its  equator,  and  entirely  detached  from  the 
body  of  the  planet.  (See  Fig.  93.)     This  ring  sometimes  casts  a 
shadow  upon  the  planet,  and  is,  in  turn,  at  times  partially  obscured 


198         OF  THE  PLANETS  AND  THEIR  PHENOMENA. 

by  the  shadow  of  the  planet ;  from  which  we  conclude  that  it  is 
opake,  and  receives  its  light  from  the  sun. 

Fig.  93.  It  is   inclined  to    the 

plane  of  the  ecliptic  in  an 
angle  of  about  28°,  and 
during  the  motion  of  Sat- 
urn in  its  orbit  it  remains 
continually  parallel  to  it- 
self. The  face  of  the  ring 
is,  therefore,  never  viewed 
perpendicularly  from  the 
earth,  and  for  this  reason 
never  appears  circular,  al- 
though such  is  its  actual 
form.  Its  apparent  form 
is  that  of  an  ellipse,  more 
or  less  eccentric,  accord- 
ing to  the  obliquity  under  which  it  is  viewed,  which  varies  with 
the  position  of  Saturn  in  its  orbit.  When  it  is  seen  under  the 
larger  angles  of  obliquity,  it  appears  as  a  luminous  band  nearly 
encircling  the  planet,  and  is  visible  in  telescopes  of  small  power. 
Stars  can, also  be  seen  between  it  and  the  planet  in  these  positions. 
At  other  times,  when  viewed  very  obliquely,  it  can  be  seen  only 
with  telescopes  of  high  power.  When  it  is  approaching  the  latter 
state,  it  has  the  appearance  of  two  handles  or  ansce,  one  on  each 
side  of  the  planet. 

It  is  also  at  times  invisible.  This  is  the  case  whenever  the 
earth  and  sun  are  on  different  sides  of  the  plane  of  the  ring,  for 
the  reason  that  the  illuminated  face  is  then  turned  from  the  earth. 
When  the  plane  of  the  ring  passes  through  the  centre  of  the  sun, 
the  illuminated  edge  can  be  seen  only  in  telescopes  of  extraordi- 
nary power,  and  appears  as  a  thread  of  light  cutting  the  disc  of 
the  planet 

535.  Since  the  orbit  of  Saturn  is  very  large  in  comparison  with 
the  orbit  of  the  earth,  the  plane  of  the  ring,  during  the  greater 
part  of  the  revolution  of  Saturn,  will  pass  without  the  orbit  of  the 
earth ;  and  when  this  is  the  case  the  ring  will  be  visible,  as  the 
earth  and  sun  will  be  on  the  same  side  of  its  plane.  During  the 
period,  which  is  about  a  ye'ar,  that  the  plane  of  the  ring  is  passing 
by  the  orbit  of  the  earth,  the  earth  will  sometimes  be  on  the  same 
side  of  it  as  the  sun,  and  sometimes  on  opposite  sides.  In  the 
latter  case  the  ring  will  be  invisible,  and  in  the  former  will  be  seen 
so  obliquely  as  to  be  visible  only  in  telescopes  of  considerable  or 
great  power.  All  this  will  perhaps  be  better  understood  on  con- 
sulting Fig.  94,  where  efg  represents  the  orbit  of  the  earth.  The 
appearances  of  the  ring  in  the  different  positions  of  the  planet  in  it| 
orbit  are  delineated  in  the  figure 

The  plane  of  the  ring  will  pass  through  the  sun  every  semi- 


SATURN  S    RING, 


199 


revolution  of  Saturn,  or,  at  a  mean,  about  every  15  years,  and  at 
the  epochs  at  which  the  longitude  of  the  planet  is  respectively 
170°  and  350°.  The  ring  will  then  disappear  once  in  about  15 
years  ;  but,  owing  to  the  different  situations  of  the  earth  in  its  or- 

Fig.  94. 


bit,  under  circumstances  oftentimes  quite  different.  And  the  dis- 
appearance will  occur  when  the  longitude  of  the  planet  is  about 
170°,  or  350°.  The  ring  will  be  seen  to  the  greatest  advantage 
when  the  longitude  of  the  planet  is  not  far  from  80°  or  260°. 
The  last  disappearance  took  place  in  1833 ;  the  next  will  be  in  1847. 
At  the  present  time  (1845)  the  north  face  of  the  ring  is  visible. 

536.  From  observations  made  upon  bright  spots  seen  on  the 
face  of  the  ring,  Herschel  discovered  that  it  revolved  from  west  to 
east  about  an  axis  perpendicular  to  its  plane,  and  passing  through 
the  centre  of  the  planet,  (or  very  nearly.)     The  period  of  its  ro- 
tation is  lOh.  32m.     It  is  remarkable  that  this  is  the  period  in 
which  a  satellite  assumed  to  be  at  a  mean  distance  equal  to  the 
mean  distance  of  the  particles  of  the  ring,  would  revolve  around 
the  primary  according  to  the  third  law  of  Kepler. 

The  breadth  of  the  ring  is  about  one-half  greater  than  its  dis- 
tance from  the  surface  of  the  planet,  and  is  about  equal  to  one- 
third  the  diameter  of  the  planet,  or  29,000  miles. 

537.  What  we  have  called  Saturn's  ring  consists  in  fact  of  two 
concentric  rings,  which  turn  together,  although  entirely  detached 
from  each  other.     The  void  space  between  them  is  perceived  in 
telescopes' of  high  power,  under  the  form  of  a  black  oval  line. 
According  to  the  calculations  of  Sir  John  Herschel,  from  the  mi- 
crometric  measures  of  Professor  Struve,  the  breadth  of  the  interior 
ring  is  about  17,200  miles,  and  of  the  exterior  about  10,600  miles; 
the  interval  between  the  rings  is  nearly  1800  miles,  and  the  dis- 
tance from  the  planet  to  the  inside  of  the  interior  ring  is  a  little 
over  19,000  miles.     The  thickness  of  the  rings  is  not  well  known.; 
the  edge  subtends  an  angle  much  less  than  1",  which,  at  the  dis- 
tance of  the  planet,  answers  to  about  5000  miles.     Herschel  makes 
it  less  than  250  miles.     (See  Note  X.) 

538.  Professor  Bessel  has  shown  that  the  double  ring  is  not  bounded  by  parallel 
plane  surfaces.     He  infers  this  to  be  the  case  from  the  fact  that  at  almost  every 


200  OF  THE  PLANETS  AND  THEIR  PHENOMENA. 

disappearance  or  reappearance  of  the  ring,  the  two  ansse  have  not  disappeared  01 
reappeared  at  the  same  time.  He  has  also  found,  from  a  discussion  of  the  obser- 
vations which  have  been  made  upon  the  disappearances  and  reappearances  of  the 
ring,  that  they  cannot  be  satisfied  by  supposing  the  two  faces  of  the  ring  to  be 
parallel  planes.  In  view  of  all  the  facts,  it  seems  most  probable  that  the  cross  sec- 
tion of  each  ring  is  a  very  eccentric  ellipse,  instead  of  a  rectangle,  and  that  it  varies 
somewhat  in  size  from  one  part  of  the  ring  to  another.  It  may  have  irregularities 
on  its  surface  as  great  or  greater  than  those  which  diversify  the  surface  of  the  earth. 

539.  Whatever  may  be  the  form  of  the  rings,  their  matter  is  not  uniformly  dis- 
tributed.    For  recent  micrometric  measurements  of  great  delicacy,  made  by  Pro- 
fessor Struve,  have  made  known  the  fact,  that  the  rings  are  not  concentric  with 
the  planet,  but  that  their  centre  of  gravity  revolves  in  a  minute  orbit  about  the 
centre  of  the  planet.     Laplace  had  previously  inferred,  from  the  principle  of  gravi- 
tation, that  this  circumstance  was  essential  to  the  stability  of  the  rings.     He  de- 
monstrated that  if  the  centre  of  gravity  of  either  ring  were  once  strictly  coincident 
with  the  centre  of  gravity  of  the  planet,  the  slightest  disturbing  force,  such  as  the 
attraction  of  a  satellite,  would  destroy  the  equilibrium  of  the  ring,  and  eventually 
cause  the  ring  to  precipitate  itself  upon  the  planet. 

540.  In  respect  to  the  origin  of  Saturn's  ring,  Sir  John  Herschel  has  offered  the 
interesting  suggestion,  that,  as  the  smallest  difference  of  velocity  in  space  between 
the  planet  and  ring  must  infallibly  precipitate  the  latter  on  the  former,  never  more 
to  separate,  it  follows  either  that  their  motions  in  their  common  orbit  around  the 
sun  must  have  been  adjusted  by  an  external  power  with  the  minutest  precision, 
or  that  the  ring  must  have  been  formed  about  the  planet  while  subject  to  their 
common  orbitual  motion,  and  under  the  full  and  free  influence  of  all  the  acting 
forces.     The  latter  supposition  accords  with  Laplace's  theory  of  the  progressive 
creation  of  the  universe,  hereafter  to  be  noticed. 

541.  The  satellites  of  Saturn  were  discovered,  the  6th  in  the 
order  of  distance  by  Huygens,  in  1655,  with  a  telescope  of  12  feet 
focus ;  the  3d,  4th,  5th,  and  8th,  by  Dominique  Cassini,  between 
the  years  1670  and  1685,  with  refracting  telescopes  of  100  and 
136  feet  in  length  ;  and  the  1st  and  2d  by  Sir  William  Herschel, 
in  1789,  with  his  great  reflecting  telescope  of  40  feet  focus.     All 
but  the  1st  and  2d  are  visible  in  a  telescope  of  a  large  aperture, 
with  a  magnifying  power  of  200.     (See  Note  XL) 

They  all,  with  the  exception  of  the  8th,  revolve  very  nearly  in 
the  plane  of  the  ring  and  of  the  equator  of  the  primary.  The  or- 
bit of  the  8th  is  inclined  under  a  considerable  angle  to  this  plane. 
According  to  Sir  John  Herschel,  the  6th  satellite  is  much  the  lar- 
gest, and  is  estimated  to  be  not  much  inferior  to  Mars  in  size.  The 
others  diminish  in  size  as  we  proceed  inward ;  until  the  1  st  and  2d 
are  so  small,  and  so  near  the  ring,  that  they  have  never  been  dis- 
cerned but  with  the  most  powerful  telescopes  which  have  yet  been 
constructed  ;  and  with  these  only  at  the  time  of  the  disappearance 
of  the  ring,  (to  ordinary  telescopes,)  when  they  have  been  seen  as 
minute  points  of  light  skirting  the  narrow  line  of  the  luminous 
edge  of  the  ring. 

The  8th  satellite  is  subject  to  periodical  variations  of  lustre, 
which  prove  its  rotation  on  an  axis  in  the  period  of  a  sidereal  revo- 
lution of  Saturn. 

URANUS  AND  ITS  SATELLITES. 

642.  Uranus  is  scarcely  visible  to  the  naked  eye.  In  a  tele- 
scope it  appears  as  a  small  round  uniformly  illuminated  disc.  Its 


URANUS — VESTA — JUNO CERES — PALLAS.        201 

apparent  diameter  is  about  4",  from  which  it  never  varies  much, 
owing  to  the  smallness  of  the  earth's  orbit  in  comparison  with  its 
own.  Its  real  diameter  is  about  34,500  miles,  and  its  bulk  82 
times  that  of  the  earth.  Analogy  leads  us  to  believe  that  this  pla- 
net is  opake  and  turns  on  an  axis,  but  there  is  no  direct  proof  that 
this  is  the  case. 

543.  The  satellites  of  Uranus  were  discovered  by  Sir  W.  Her- 
schel.     They  are  discernible  only  with  telescopes  of  the  highest 
power.     (See  Note  XII.) 

VESTA JUNO CERES PALLAS. 

544.  These  four  planets,  although  less  distant  than  several  of 
the  others,  are  so  extremely  small,  that  they  cannot  be  seen  with- 
out the  aid  of  a  telescope. 

Vesta  is  the  most  brilliant,  and  shines  with  a  white  light.  In 
the  telescope  it  appears  as  a  star  of  about  the  6th  magnitude.  Juno 
and  Ceres  have  the  apparent  size  of  a  star  of  the  8th  magnitude ; 
and  together  with  Pallas  have  a  ruddy  aspect  and  a  variable  lus- 
tre, indicative  of  the  presence  of  atmospheres  of  variable  density 
and  purity.  Ceres  and  Pallas  generally  shine  with  a  pale  dull 
light,  and  are  seen  surrounded  with  a  nebulosity,  or  haziness  of, 
according  to  Herschel,  from  three  to  six  times  the  extent  of  the 
body  of  the  planet.  This  haziness  is  sometimes  so  decided  as  to 
conceal  the  body  of  the  planet  from  view,  and  at  other  times  en- 
tirely disappears,  leaving  the  disc  of  the  planet  sharply  denned  and 
alone  visible. 

545.  The  actual  magnitudes  of  these  planets  are  not  well  known. 
The  determinations  of  different  Astronomers  are  widely  different. 
The  following  are  perhaps  the  nearest  approximations  to  their  true 
diameters  that  have  yet  been  obtained  :  Vesta  270  miles ;  Juno 
460  miles  ;  Ceres  460  miles  ;  Pallas  670  miles. 


CHAPTER  XVII. 

OF    COMETS. 
THEIR  GENERAL  APPEARANCE— VARIETIES  OF  APPEARANCE. 

546.  THE  general  appearance  of  comets  is  that  of  a  mass  of 
some  luminous  nebulous  substance,  to  which  the  name  Coma  has 
been  given,  condensed  towards  its  centre  around  a  brilliant  Nucleus 
that  is  in  general  not  very  distinctly  denned,  from  which  proceeds 
in  a  direction  opposite  to  the  sun  a  fainter  stream  or  train  of  simi- 
lar nebulous  matter,  called  the  Tail.  The  coma  and  nucleus  to- 
gether form  what  is  called  the  Head  of  the  Comet.  (See  Fig.  95.) 

26 


202  OF   COMETS. 

The  tail  gradually  increases  in  width,  and  at  the  same  time  di- 
minishes in  distinctness  from  the  head  to  its  extremity,  where  it  is 
generally  many  times  wider  than  at  the  head,  and  fades  away  un- 

Fig.  95. 


Great  Comet  of  18 11. 

til  it  is  lost  in  the  general  light  of  the  sky.  It  is,  in  general,  less 
bright  along  its  middle  than  at  the  borders.  From  this  cause  the  tail 
sometimes  seems  to  be  divided,  along  a  greater  or  Less  portion  of 
its  length,  into  two  separate  tails  or  streams  of  light,  with  a  com- 
parative dark  space  between  them.  Ordinarily  it  is  not  straight, 
that  is,  coincident  with  a  great  circle  of  the  heavens,  but  concave 
towards  that  part  of  the  heavens  which  the  comet  has  just  left. 
This  curvature  of  the  tail  is  most  observable  near  its  extremity. 
The  most  remarkable  example  is  that  of  the  comet  of  1744,  which 
was  bent  so  as  to  form  nearly  a  quarter  of  a  circle.  Nor  does  the 
general  direction  of  the  tail  usually  coincide  exactly  with  the  great 
circle  passing  through  the  sun  and  the  head  of  the  comet,  but  de- 
viates more  or  less  from  this,  the  position  of  exact  opposition  to  the 
sun 'in  the  heavens,  on  the  side  towards  the  quarter  of  the  heavens 
just  traversed  by  the  comet.  This  deviation  is  quite  different  for 
different  comets,  and  varies  materially  for  the  same  comet  while  it 
continues  visible.  It  has  even  amounted  in  some  instances  to  a 
right  angle. 

547.  The  apparent  length  of 'the  tail  varies  from  one  comet  to 
another  from  zero  to  100°  and  more  ;  and  ordinarily  the  tail  of  the 
same  comet  increases  and  diminishes  very  much  in  length  during 


GENERAL  APPEARANCE  OF  COMETS. 


203 


the  period  of  its  visibility.  When  a  comet  first  appears,  in  general, 
no  tail  is  perceptible,  and  its  light  is  very  faint.  As  it  approaches 
the  sun,  it  becomes  brighter:  the  tail  also  after  a  time  shoots  out 
from  the  coma,  and  increases  from  day  10  day  in  extent  and  dis- 
tinctness. As  the  comet  recedes  from  the  sun,  the  tail  precedes 
the  head,  being  still  on  the  opposite  side  from  the  sun,  and  grows 
less  and  less  at  the  same  time  that,  along  with  the  head,  it  de- 
creases in  brightness,  till  at  length  the  comet  resumes  nearly  its 
first  appearance,  and  ^finally  disappears.  (See  Fig.  97.)  It  some- 
times happens  that,  owing  to  peculiar  circumstances,  Fig.  96. 
a  comet  does  not  make  its  appearance  in  the  firma- 
ment until  after  it  has  passed  the  sun  in  the  heavens, 
and  not  until  it  has  attained  to  more  or  less  distinct- 
ness, and  is  furnished  with  a  tail  of  considerable  or  \ 
even  great  length.  This  was  remarkably  the  case  i 
with  the  great  comet,  of  1843.  (See  Art.  326  ;  also 
Fig.  96.) 

548.  The  tail  of  a  comet  is  the  longest,  and  the 
whole  comet  is  intrinsically  the  most  luminous,  not 
long  after  it  has  passed  its  perihelion.     Its  apparent 
size  and  lustre  will  not,  however,  necessarily  be  the  1 
greatest  at  this  time,  as  they  will  depend  upon  the  T 
distance  and  position  of  the  earth,  as  well  as  the  ac- 
tual size  and  intrinsic  brightness  of  the  comet.     To 
Fig.  97. 


N 


illustrate  this,  let  abed  (Fig.  97)  represent  the  orbit  j 
of  the  earth,  and  MPN  the  orbit  of  a  comet,  having  j 
its  perihelion  at  P.    Now,  if  the  earth  should  chance 
to'  be  at  a  when  the  comet,  moving  towards  its  peri- 
helion, is  at  r,  it  might  very  well  happen  that  the 
comet  would  appear  larger  and  more  distinct  than 


204  OF    COMETS. 

when  it  had  reached  the  more  remote  point  s,  although  when  at  the 
latter  point  it  would  in  reality  be  larger  and  brighter  than  when  at 
r.  It  would  be  the  most  conspicuous  possible  if  the  earth  should 
be  in  the  vicinity  of  c  or  b  soon  after  the  perihelion  passage  :  and 
it  would  be  the  least  conspicuous  possible  if  the  comet,  sup- 
posed to  be  moving  in  the  direction  NPM,  should  pass  from  N 
around  to  M,  while  the  earth  is  moving  around  from  a  to  b  or  c,  so 
as  to  be  continually  comparatively  remote  from  the  comet,  and  so 
that  the  comet  will  be  in  conjunction  with  the  sun  at  the  time  after 
the  perihelion  passage  when  its  actual  size  and  intrinsic  lustre  are 
the  greatest.  It  is  to  be  observed  that  the  apparent  lustre  of  a 
comet  is  sometimes  very  much  enhanced  by  the  great  obliquity  of 
the  tail,  in  some  of  its  positions,  to  the  line  of  sight.  This  seems 
to  have  been  the  case  with  the  comet  of  1843,  on  February  28th, 
(see  Fig.  56,)  and  was  doubtless  one  reason  of  its  being  so  very 
bright  as  to  be  seen  in  open  day  in  the  immediate  vicinity  of  the 
sun. 

Since  the  earth  may  have  every  variety  of  position  in  its  orbit 
at  the  different  returns  of  the  same  comet  to  its  perihelion,  it  will 
be  seen,  on  examining  Fig.  97,  that  the  circumstances  of  the  ap- 
pearance and  disappearance  of  the  comet,  as  well  as  its  size  and 
distinctness,  may  be  very  various  at  its  different  returns.  This 
has  been  strikingly  true  in  the  case  of  Halley's  Comet.  Gambart's 
Comet  was  also  invisible  in  its  return  to  its  perihelion  in  1839,  by 
reason  of  its  continual  proximity  to  the  line  of  direction  of  the  sun 
as  seen  from  the  earth,  and  its  great  distance  from  the  earth. 

549.  Individual  comets  offer  considerable  varieties  of  aspect. 
Some  comets  have  been  seen  which  were  wholly  destitute  of  a 
tail :  such,  among  others,  was  the  comet  of  1682,  which  Cassini 
describes  as  being  as  round  and  as  bright  as  Jupiter.    Others  have 
had  more  than  one  luminous  train.     The  comet  of  1744  was  pro- 
vided with  six,  which  were   spread  out,  like  an  immense  fan, 
through  an  angle  of  1 17°  ;  and  that  of  1823  with  two,  one  directed 
from  the  sun  in  the  heavens,  and,  what  is  very  remarkable,  another 
smaller  and  fainter  one  directed  towards  the  sun.  Others  still  have 
had  no  perceptible  nucleus,  as  the  comets  of  1795  and  1804. 

The  comets  that  are  visible  only  in  telescopes,  which  are  very 
numerous,  have,  generally,  no  distinct  nucleus,  and  are  often  entire- 
ly destitute  of  every  vestige  of  a  tail.  They  have  the  appearance  of 
round  masses  of  luminous  vapor,  somewhat  more  dense  towards  the 
centre.  Such  are  Encke's  and  Biela's  comets.  (See  Fig.  98.)  The 
point  of  greatest  condensation  is  often  more  or  less  removed  from  the 
centre  of  figure  on  the  side  towards  the  sun ;  and  sometimes  also 
on  the  opposite  side.  (See  Note  XIII.) 

550.  The  comets  which  have  had  the  longest  tails  are  those  of 
1680,  1769,  and  1618.  The  tail  of  the  great  comet  of  1680,  when 
apparently  the  longest,  extended  to  a  distance  of  70°  from  the  head  • 
that  of  the  comet  of  1 769,  a  distance  of  97°  ;  and  that  of  the  com- 


FORM,  STRUCTURE,  AND  DIMENSIONS  OF  COMETS.  205 

et  of  1618,  104°.  These  are  the  apparent  lengths  as  seen  at  cer- 
tain places.  By  reason  of  the  different  degrees  of  purity  and  den- 
sity of  the  air  through  which  it  is  seen,  the  tail  of  the  same  comet 
often  appears  of  a  very  different  length  to  observers  at  different 

Fig.  98. 


Encke's  Comet. 

places.  Thus,  the  comet  of  1769,  which  at  the  Isle  of  Bourbon 
seemed  to  have  a  tail  of  97°  in  length,  at  Paris  was  seen  with  a  tail 
of  only  60°.  From  this  general  fact  we  may  infer  that  the  actual 
tail  extends  an  unknown  distance  beyond  the  extremity  of  the  ap- 
parent tail. 

FORM,  STRUCTURE,  AND  DIMENSIONS  OF  COMETS. 

551.  The  general  form  and  structure  of  comets,  so  far  as  they 
can  be  ascertained  from  the  study  of  the  details  of  their  appear- 
ance, may  be  described  as  follows  :  The  head  of  a  comet  consists 
of  a  central  nucleus,  or  mass  of  matter  brighter  and  denser  than 
the  other  portions  of  the  comet,  enveloped  on  the  side  towards 
the  sun,  and  ordinarily  at  a  great  distance  from  its  surface  in 
comparison  with  its  own  dimensions,  by  a  globular  nebulous 
mass  of  great  thickness,  called  the  Nebulosity,  or  nebulous  En- 
velope. This,  it  is  said,  never  completely  surrounds  the  nu- 
cleus, except  in  the  case  of  comets  which  have  no  tails.  It  forms 
a  sort  of  hemispherical  cap  to  the  nucleus  on  the  side  towards  the 
sun.  Its  form,  j^owever,  is  not  truly  spherical,  but  approximates 
to  that  of  an  hyperboloid  having  the  nucleus  in  its  focus  and  its  ver- 
tex turned  towards  the  sun.  The  tail  begins  where  the  nebulosity 
terminates,  and  seems,  in  general,  to  be  merely  the  continuation  01 
this  in  nearly  a  straight  line  beyond  the  nucleus.  There  is  ordina- 
rily, as  has  been  already  intimated,  a  distinct  space  containing  but 
little  luminous  matter  between  the  nucleus  and  the  nebulosity,  but 
this  is  not  always  the  case.  The  tail  of  a  comet  has  the  shape  of 
a  hollow  truncated  cone,  with  its  smaller  base  in  the  nebulosity  of 
the  head ;  with  this  difference,  however,  that  the  sides  are  usually 


206  OF    COMETS. 

more  or  less  curved,  and  ordinarily  concave  towards  the  axis.  Tha' 
the  tail  is  hollow  is  evident  from  the  fact,  already  noticed,  that  on 
whichever  side  it  is  viewed  it  appears  less  bright  along  the  middle 
than  at  the  borders.  There  can  be  less  luminous  matter  on  a  line 
of  sight  passing  through  the  middle,  than  on  one  passing  near  one 
of  the  edges,  only  on  the  supposition  that  the  tail  is  hollow.  The 
whole  tail  is  generally  bent  so  as  to  be  concave  towards  the  regions 
of  space  which  the  comet  has  just  left. 

552.  In  some  instances  the  nucleus  is  furnished  with  several 
envelopes  concentric  with  it :  which  are  formed  in  succession  as 
the  comet  approaches  the  sun.     For  example,  the  comet  of  1744, 
eight  days  after  the  perihelion  passage,  had  three  envelopes.  Some- 
times each  of  them  is  provided  with  a  tail.     Each  of  tfeese  sev- 
eral tails  lying  one  within  the  other,  being  hollow,  may  in  conse- 
quence appear  so  faint  along  its  middle  as  to  have  the  aspect  of 
two  distinct  tails.  A.  comet  which  has  in  reality  three  separate  tails, 
might  thus  appear  to  be  supplied  with  six,  as  was  the  comet  of 
1744.    If  the  different  envelopes  were  not  distinctly  separate  from 
each  other,  then  we  should  have  all  the  tails  appearing  to  proceed 
from  the  same  nebulous  mass. 

553.  Supernumerary  tails,  shorter  and  less   distinct  than  the 
principal  tail,  are  by  no  means  uncommon ;  but  they  generally 
appear  quite  suddenly,  and  as  suddenly  disappear  in  a  few  days,  as 
if  the  stock  of  materials  from  which  they  were  supplied  had  be- 
come  exhausted.      These    secondary   tails,   by   their   periodical 
changes  of  position  from  the  one  side  of  the  principal  tail  to  the 
other,  have  made  known  the  fact  that  the  comets  to  which  they 
belonged  had  a  rotatory  motion  around  the  axis  or  central  line  of.  the 
tail.     The  same  fact  has  been  inferred  from  other  phenomena,  in 
the  case  of  some  other  comets,  as  the  great  comet  of  1811,  and 
Halley's  comet  in  1835. 

554.  The  general  position  of  the  tail  of  a  comet  is  nearly  but 
Hot  exactly  in  the  prolongation  of  the  line  of  the  centres  of  the 
sun  and  head  of  the  comet,  or  of  the  radius-vector  of  the  comet. 
•(See  Fig.  97.)   It  deviates  from  this  line  on  the  side  of  the  regions 
of  space  which  the  comet  has  just  left ;  and  the  angle  of  deviation, 
'Which,  when  the  comet  is  first  seen  at  a  distance  from  the  sun,  is 
very  small  or  not  at  all  perceptible,  increases  as  the  comet  a 


'proaches  the  sun,  and  attains  to  its  maximum  vakie  soon  after  the 
perihelion  passage  ;  after  which  it  decreases,  and  finally,  at  a  dis- 
tance from  the  sun,  becomes  insensible.  For  example,  the  angle 
of  deviation  of  the  tail  of  the  great  comet  of  1811  attained  tO'ite 
maximum  about  ten  days  after  the  perihelion  passage,  and  was 
then  about  1 1°.  In  the  case  of  the  comet  of  1664,  the  same  angle 
about  two  weeks  after  the  perihelion  passage  was  43°,  and  was 
then,  decreasing  at  the  rate  of  8°  per  day..  ^ 

The  comet  of  1823  might  seem  to  (present  aat  »exoeption  to  the 
general  fact  that  the  tail  of  a  comet  is  nearly  opposite  to  the  sun ; 


PHYSICAL  CONSTITUTION  OF  COMETS.  207 

but  Arago  has  suggested  that  the  probable  cause  of  the  singular 
phenomenon  of  a  secondary  tail,  apparently  directed  towards  the 
sun  in  the  heavens,  was  that  the  earth  was  in  such  a  position  that 
the  two  tails,  although  in  fact  inclined  to  each  other  under  a  small 
angle,  were  directed  towards  different  sides  of  the  earth,  and  thus 
were  referred  to  the  heavens  so  as  to  appear  nearly  opposite. 

The  same  principle  will  serve  to  show  that  the  deviation  of  the 
tail  of  a  comet,  from  the  position  of  exact  opposition  to  the  sun, 
may  appear  to  be  much  greater  than  it  actually  is,  by  reason  of  the 
earth  happening  to  be  within  the  angle  formed  by  the  direction  of 
the  tail  with  the  radius-vector  prolonged. 

555.  Comets  are  the  most  voluminous  bodies  in  the  solar  sys- 
tem.    The  tail  of  the  great  comet  of  1680  was  found  by  Newton 
to  have  been,  when  longest,  no  less  than  123,000,000  miles  in 
length :  according  to  Professor  Peirce,  the  remarkable  comet  of 
1843,  about  three  weeks  after  its  perihelion  passage,  had  a  tail  of 
over  200,000,000*  miles  in  length.     Other  comets  have  had  tails 
of  from  fifty  to  a  hundred  millions  of  miles  in  length.     The  heads 
of  comets  are  usually  many  thousand  miles  in  diameter.     That  of 
the  comet  of  181 1  had  a  diameter  of  132,000  miles.     Its  envelope 
or  nebulosity  was  30,000  miles  in  thickness  ;  and  the  inner  surface 
of  this  was  *io  less  than  36,000  miles  distant  from  the  centre  of  the 
nucleus.     The  head  of  the  great  comet  of  1843  was  about  30,000 
miles  in  diameter. 

The  nuclei  of  comets  are  in  general  only  a  few  hundred  miles  in 
diameter  :  but  according  to  Schroeter  the  nucleus  of  the  comet  of 
1811  had  a  diameter  of  2600  miles ;  and  the  nucleus  of  the  comet  of 
1843  seems  to  have  been  still  greater.  On  the  other  hand,  the 
comet  of  1798  had  a  nucleus  of  less  than  50  miles  in  diameter. 

It  is  important  to  observe  that  the  dimensions  of  comets  are  sub- 
ject to  continual  variations.  The  tail  increases  as  the  comet  ap- 
proaches the  sun,  and  attains  to  its  greatest  size  a  certain  time  after 
the  perihelion  passage  ;  after  which  it  decreases.  The  head,  on 
the  contrary,  generally  diminishes  in  size  during  the  approach  to 
the  sun,  and  augments  during  the  recess  from  him.  The  changes 
are  often  very  sudden  and  rapid. 

PHYSICAL  CONSTITUTION  OF  COMETS. 

556.  The  quantity  of  matter  which  enters  into  the  constitution 
of  a  comet  is  exceedingly  small.     This  is  proved  by  the  fact  that 
the  comets  have  had  no  influence  upon  the  motions  of  the  planets  or 
satellites,  although  they  have  in  many  instances  passed  near,  thegj'e 
bodies.     The  comet  of  1770,  which  was  quite  large  and  bright, 
passed  through  the  midst  of  Jupiter's  satellites,  without  deranging 
their  motions  in  the  least  perceptible  degree.    Moreover,  since  this 
small  quantity  of  matter  is  dispersed  over  a  space  of  tens  of  thou- 

*  According  to  later  determinations  108,000,000  miles. 


208  OF  COMETS. 

sands,  or  millions  of  miles  (if  we  include  the  tail,)  in  linear  extent, 
the  nebulous  matter  of  comets  must  be  incalculably  less  dense  than 
the  solid  matter  of  the  planets.  In  fact,  the  cometic  matter,  with 
the  exception  perhaps  of  that  of  the  nucleus,  is  inconceivably  more 
rare  and  subtile  than  the  lightest  known  gas,  or  the  most  evanescent 
film  of  vapor  that  ever  makes  its  appearance  in  our  sky  ;  for  faint 
telescopic  stars  are  distinctly  visible  through  all  parts  of  the  comet, 
with,  it  may  be,  the  exception  of  the  nucleus  in  some  instances, 
notwithstanding  the  great  space  occupied  by  the  matter  of  the 
comet  which  the  light  of  the  star  has  to  traverse.  The  matter  of 
the  tail  of  a  comet  is  even  more  attenuated  than  that  of  the  general 
mass  of  the  nebulosity  of  the  head  ;  but  is  apparently  of  the  same 
nature,  and  derived  from  the  head.  The  nucleus  is  supposed  by 
some  astronomers  to  be,  in  some  instances,  a  solid,  partially 
or  wholly  convertible  into  vapor,  under  the  influence  of  the  sun  ; 
by  others,  to  be  in  all  cases  the  same  species  of  matter  as  is  in  the 
nebulosity,  only  in  a  more  condensed  state  ;  and  by  others  still,  to 
be  a  solid  of  permanent  dimensions,  with  a  thick  stratum  of  con- 
densed vapors  resting  upon  its  surface.  Whichever  of  these  views 
be  adopted,  it  is  a  matter  of  observation  that  the  nebulosity  fre- 
quently receives  fresh  supplies  of  nebulous  matter  from  the  nu- 
cleus. It  was  the  opinion  of  Sir  William  Herschel,  an<J  it  has  been 
the  more  generally  received  notion  since  his  time,  that  the  nucleus 
of  a  comet  is  surrounded  with  a  transparent  atmosphere  of  vast 
extent,  within  which  the  nebulous  envelope  floats,  as  do  clouds  in 
the  earth's  atmosphere.  But  Olbers,  and  after  him  Bessel,  con- 
ceives the  nebulous  matter  of  the  head  to  be  either  in  the  act  of 
flowing  away  into  the  tail  under  the  influence  of  a  repulsion  from 
the  nucleus  and  the  sun,  or  in  a  state  of  equilibrium  under  the  ac- 
tion of  these  forces  and  the  attraction  of  the  nucleus. 

It  is  not  yet  definitively  settled  whether  the  cometic  matter  is 
self-luminous,  or  shines  with  the  light  received  from  the  sun ;  but 
it  is  the  general  opinion  that  it  derives  its  light  from  the  sun. 

CONSTITUTION  AND  MODE  OF  FORMATION  OF  THE  TAILS  OF  COMETS. 

557.  Upon  this  topic  we  may  lay  down  the  following  postulates.  1.  The  gen- 
eral  situation  of  the  tail  of  a  comet  with  respect  to  the  sun,  shows  that  the  sun  is 
concerned,  either  directly  or  indirectly,  in  its  formation.  The  changes  which  take 
place  in  the  dimensions  of  a  comet,  both  in  approaching  the  sun  and  receding  from 
him,  conduct  to  the  same  inference.  2.  Since  the  tail  lies  in  the  direction  of  the 
radius-vector  prolonged  beyond  the  head,  the  particles  of  matter  of  which  it  is 
made  up  must  have  been  driven  off  by  some  force  exerted  in  a  direction  from  the 
sun.  3.  This  force  cannot  emanate  from  the  nucleus,  for  such  a  force  would  ex- 
pel the  nebulous  matter  surrounding  the  nucleus  in  all  directions,  instead  of  one 
direction  only.  It  is,  however,  conceivable  that,  as  Olbers  supposes,  the  nebulous 
matter  is  in  the  first  instance  expelled  from  the  nucleus  by  its  repulsive  action, 
taking  effect  chiefly  on  the  side  towards  the  sun,  and  afterwards  driven  past  the 
nucleus  into  the  tail  by  a  repulsion  from  the  sun.  4.  There  seems,  then,  to  be  little 
room  to  doubt  that  the  matter  of  the  tail  is  driven  off  from  the  head  by  some  force 
foreign  to  the  comet,  and  taking  effect  from  the  sun  outwards.  5.  This  force, 


FORMATION  OF  THE  TAILS  OF  COMETS.  209 

whatever  may  be  its  nature,  extends  far  beyond  the  earth's  orbit.  For  comets 
have  been  seen  provided  with  tails  of  great  length,  though  their  perihelion  distance 
exceeded  the  radius  of  the  earth's  orbit,  (e.  g.  the  great  comet  of  1811.)  Nothing 
can  be  predicated  with  certainty  with  respect  to  the  law  of  variation  of  this  force, 
but  it  is  at  least  probable  that,  like  all  known  central  forces,  it  varies  inversely  as 
the  square  of  the  distance. 

558.  Whatever  may  be  the  nature  of  the  force  in  question ;  whether  it  consists 
in  an  impulsive  action  of  the  sun's  rays,  as  Euler  imagined,  or  in  a  repulsion  by 
the  distant  mass  of  the  sun,  consequent  upon  a  polarity  of  the  cometic  particles 
induced  by  some  action  of  the  sun,  as  supposed  by  Olbers  and  Bessel,  we  will  call 
it  the  repulsive  force  of  the  sun.  Granting  its  existence,  there  are  two  modes  in 
which  we  may  conceive  it  to  operate  in  forming  the  tail.  We  may  suppose  that 
it  drives  off  the  nebulous  matter  to  greater  and  greater  distances,  as  its  intensity 
increases,  without  destroying  the  original  physical  connection  of  the  parts  ;  so  that 
the  tail  and  the  head  will  always  be  revolving  as  one  connected  mass.  Or  we  may 
conceive  that  it  is  continually  detaching  portions  of  the  nebulosity,  or  turning  them 
back  if  repelled  by  the  nucleus,  and  repelling  them  to  an  indefinite  distance  into 
free  space.  The  first  mentioned  conception  is  the  theory  which  has  generally  pre- 
vailed hitherto;  but  there  seem  to  be  good  and  sufficient  reasons  for  rejecting  it, 
and  adopting  the  other  in  its  stead.  1.  There  appears  to  be  no  satisfactory  rea- 
son to  be  assigned  why  the  force  which  expels  the  nebulous  matter  to  the  end  of 
the  apparent  tail  should  not  urge  it  still  farther ;  since  the  extremities  of  the  tails 
of  some  comets  are  not  so  far  removed  from  the  sun  as  the  heads  of  others,  from 
which  the  nebulous  matter  is  expelled  by  the  same  force.  To  account  for  the  sup- 
posed limited  extent  of  the  actual  tail,  we  are  forced  to  suppose  that  the  tendency 
of  the  particles  to  return  to  the  nucleus  increases  as  their  distance  from  the  nucleus 
and  from  one  another  increases ;  which  seems  highly  improbable.  2.  Bessel  has 
found  that  the  nebulous  matter  of  comets  has  no  power  to  refract  the  light  of  a 
star,  passing  through  it,  whence  he  infers  that  there  can  be  no  molecular  connec- 
tion between  the  particles.  3.  It  appears,  by  calculation,  that  in  the  case  of  the 
great  comet  of  1843,  we  cannot  find  either  in  the  repulsive  action  of  the  sun  upon 
the  tail,  or  in  the  excess  of  attraction  of  the  sun  for  the  nearer  parts  of  the  comet, 
a  force  adequate  to  keep  the  tail  continually  opposite  to  the  sun,  and  which  at  the 
same  time  will  not  sensibly  alter  the  orbit,  without  making  improbable  suppositions 
as  to  the  disproportion  between  the  quantity  of  matter  in  the  nucleus  and  tail.* 
4.  In  the  case  of  such  comets  as  that  of  1843,  and  that  of  1680,  which  come  near 
the  sun,  the  centrifugal  force  generated  by  the  great  velocity  of  rotation  about  the 
time  of  the  perihelion  passage  would  be  so  great  as  infallibly  to  dissipate  the  great- 
er  part  of  the  tail.  At  the  time  of  the  perihelion  passage  of  either  of  these  comets, 
the  centrifugal  force  must  have  exceeded  the  gravity  towards  the  nucleus  at  only 
a  few  hundred  miles  from  the  centre  of  gravity  of  the  whole  mass.  5.  Whether  we 
suppose  the  whole  mass  of  the  comet  to  be  kept  in  rotation  about  its  centre  of  grav- 
ity by  the  repulsion  or  by  the  attraction  of  the  sun,  the  velocity  of  rotation,  as  it 
is  constantly  nearly  equal  to  the  angular  velocity  of  revolution,  must  be  on  the  in- 
crease up  to  the  time  of  the  perihelion  passage.  Now,  this  will  not  undergo  any 
diminution  after  the  perihelion  passage,  as  the  action  of  the  force  would  tend  to 
increase  rather  than  diminish  it,  but  the  velocity  of  revolution  will  continually  de- 
crease :  it  follows,  therefore,  that  soon  after  the  perihelion  passage  the  velocity  of 
rotation  would  exceed  that  of  revolution,  and  continually  more  and  more ;  so  that 
ere  long  the  tail  would  inevitably  be  thrown  forward  of  the  line  of  the  radius-vector 
prolonged,  a  situation  in  which  the  tail  of  a  comet  has  never  been  seen. 

We  here  suppose  the  dimensions  of  the  comet  to  remain  the  same.  In  point  of 
fact,  the  apparent  tail  increases  in  length  for  a  certain  number  of  days  after  the 
perihelion  passage.  The  tendency  of  this  would  be  to  diminish  the  velocity  of  ro- 
tation ;  but  the  supposed  subsequent  contraction  of  the  tail  to  its  original  dimen- 
sions would  restore  the  original  velocity. 

In  view  of  all  that  has  now  been  stated,  it  seems  highly  probable  that  the  tail 
and  head  of  a  comet  do  not  form  one  connected  body  of  matter,  as  has  been  gen- 
erally supposed  ;  but,  on  the  contrary,  that  the  tail  is  made  up  of  particles  of  mat- 
ter continually  in  the  act  of  flowing  away  at  a  very  rapid  rate  from  the  head  into 

*  See  Silliman's  Journal,  vol.  xlvi,  No.  I,  page  110,  &c. 
27 


210 


OF    COMETS. 


free  space,  under  the  action  of  the  repulsive  force  of  the  sun,  so  called.  According 
to  this  view,  the  tail  which  we  see  at  any  instant  is  the  collection  of  all  the  parti- 
cles that  have  been  emitted  during  a  certain  previous  interval ;  and  at  the  end  of 
every  such  interval  we  are  looking  at  an  entirely  new  tail.  This  theory  of  the  con- 
stitution of  the  tails  of  comets  is  identical  with  Gibers' ;  but,as  we  have  seen  (556), 
Olbers  has  also  given  a  special  theory  of  the  constitution  of  the  nebulosity  of  the 
heads  of  comets,  of  which  nothing  is  here  predicated. 

559.  If  a  theory  be  true,  it  must  furnish  a  satisfactory  explanation  of  the  facts 
and  phenomena  that  fall  within  its  scope.    Let  us  examine  the  present  theory  from 
this  point  of  view.     In  the  first  place,  as  respects  the  form  of  the  tail,  it  is  mani- 
fest from  what  has  already  been  stated,  (546,)  that  the  sides  of  the  tail  must  often 
diverge  much  more  rapidly  than  the  lines  of  action  of  the  repulsive  force  of  the 
sun  upon  the  opposite  parts  of  the  head.     Calculation  shows  this  to  have  been  the 
case  even  in  the  comet  of  1843.     Now,  according  to  Olbers'  theory,  this  fact  is  a 
simple  consequence  of  the  supposed  repulsive  action  of  the  nucleus.     If  we  adopt 
Herschel's  theory  of  the  constitution  of  the  head,  we  have  apparently  a  sufficient 
cause  for  the  same  fact  in  the  centrifugal  force  generated  by  the  rotation  of  the  tail, 
(553,)  which  we  must  suppose  to  be  a  consequence  of  a  rotation  of  the  head. 

560.  In  the  next  place,  the  increase  in  the  length  of  the  apparent  tail  as  the 
comet  approaches  the  sun,  and  until  a  certain  time  after  the  perihelion  passage, 
may  be  naturally  supposed  to  proceed  from  the  emission  of  greater  quantities  of 
luminous  matter  in  a  given  time,  and  a  continued  augmentation,  up  to  the  time  of 
the  perihelion  passage,  in  the  light  received  from  the  sun.    The  actual  tail,  it  is  to 
be  observed,  is  really  indefinite  in  length,  and  terminates,  to  us,  where  its  matter 
becomes  too  much  dispersed  and  too  distant  from  the  sun,  the  probable  source  of 
its  light,  to  send  us  a  perceptible  light. 

561.  Let  us  now  see,  in  the  third  place,  how  the  theory  under  consideration  ac- 
counts for  the  situation  and  curvature  of  the  tail.    Let  PCA  (Fig.  99)  be  a  portion 
of  a  comet's  orbit,  the  sun  being  at  S  :  and  suppose  a  particle  to  be  expelled  in  the 

Fig.  99. 


direction  SAD,  when  the  head  is  at  A,  and  another  particle  to  be  driven  off  in  the 
direction  SEE,  when  the  head  is  at  B.  Each  particle  will  retain  the  orbitual  mo- 
tion which  obtained  at  the  time  of  its  departure,  as  it  moves  away  from  the  sun ; 
and  thus,  when  the  comet  has  reached  the  point  C,  instead  of  being  at  any  points 
D  and  E  on  the  lines  SAD  and  SEE,  will  be  respectively  at  certain  points  a  and  b 
farther  forward.  The  line  Cba,  which,  when  the  comet  is  at  C,  is  the  locus  of  all 
the  particles  that  have  been  emitted  during  the  interval  of  time  in  which  the  comet 
has  been  moving  over  the  arc  AC,  is  the  tail.  We  here  suppose  the  head  to  be  a 
mere  point.  If  we  conceive  the  particles  to  be  continually  emitted  from  the  mar- 
ginal parts  of  the  head,  we  shall  have  the  hollow  conical  tail  actually  observed.  It 
is  easy  to  see  that  Cba,  the  line  of  the  tail,  must  be  a  curved  line  concave  towards 


%    w 

NUMBER  AND  DISTRIBUTION  OP  THE  FIXED  STARS.  211 

*  ,  N  tx         £'* 

the  regions  of  space  which  the  comet  has  left.  Supposing  the  arc,  AC  to  be  so 
small,  or  its  curvature  to  be  so  slight  that  it  may  be  considered  as  a  straight  line, 
and  neglecting  the  change  of  the  velocity  in  the  orbit,  Ca  will  be  parallel  to  AD, 
and  Ch  parallel  to  BE,  whence  RCa  =  CSA,  and  RCb  =  CSB.  Thus  the  line 
joining  any  particle  with  the  nucleus  always  makes  an  angle  with  the  prolonga- 
tion of  the  radius-vector,  equal  to  the  motion  in  anomaly  during  the  interval 
that  has  elapsed  since  the  particle  left  the  head.  It  follows  from  this  that,  if 
we  suppose  the  velocity  of  the  particles  to  be  continually  the  same,  and  the  mo- 
tion in  anomaly  to  be  uniform,  the  deviations  of  the  particles  a  and  b  from  the 
line  of  the  radius-vector  SCR  will  be  in  the  ratio  of  the  distances  Co  and  Cb.  But, 
in  point  of  fact,  the  velocity  increases  with  the  distance,  so  that  the  curvature  of 
the  tail  will  be  less  than  on  the  supposition  just  made. 

As  to  the  amount  of  the  deviation  of  the  tail  from  the  line  of  the  radius- vector, 
it  must  depend  upon  the  proportion  between  the  velocity  of  the  particles  and  the 
velocity  of  the  head  in  its  orbit:  and  it  follows  from  the  principle  just  established, 
that  unless  the  velocities  of  emission  augment  as  rapidly  as  the  velocity  of  revolu- 
tion, the  deviation  in  question  will  increase  to  the  perihelion,  and  afterwards  de- 
crease ;  as  it  is  in  fact  known  to  do. 

562.  In  support  of  Olbers'  theory  of  a  repulsion  from  the  nucleus,  it  may  be  sta- 
ted, that  the  form  of  the  nebulosity  which  this  theory  requires,  was  found  by  obser- 
vation to  obtain  in  the  case  of  the  great  comet  of  1811,  and  also  of  Halley's  Com- 
et in  1835.* 


CHAPTER   XVIII. 

OP    THE    FIXED    STARS 
THEIR  NUMBER  AND  DISTRIBUTION  OVER  THE  HEAVENS. 

563.  THE  number  of  stars  visible  to  the  naked  eye,  in  the  entire 
sphere  of  the  heavens,  is  from  6000  to  7000 ;  of  which  nearly 
4000  are  in  the  northern  hemisphere  ;  but  not  more  than  2000  can 
be  seen  with  the  naked  eye  at  any  one  time  at  a  given  place. 
The  telescope  brings  into  view  many  millions,  and  every  material 
augmentation  of  its  space-penetrating  power  greatly  increases  the 
number. 

564.  As  to  the  number  of  stars  belonging  to  each  different  mag- 
nitude, astronomers  assign  from  20  to  24  to  the  first  magnitude, 
from  50  to  60  to  the  second,  about  200  to  the  third,  and  so  on ; 
the  numbers  increasing  very  rapidly  as  we  descend  in  the  scale 
of  brightness  ;  the  whole  number  of  stars  already  registered  down 
to  the  seventh  magnitude,  inclusive,  amounting  to  12,000  or  15,000.f 

The  reason  of  this  increase  in  the  number  of  the  stars,  as  we 
descend  from  one  magnitude  to  another,  is  undoubtedly  that  in 
general  the  stars  are  less  bright  in  proportion  as  their  distance  is 
greater ;  while  the  average  distance  between  contiguous  stars  is 
about  the  same  for  one  magnitude  as  for  another.  It  is  easy  to 
see  that  upon  these  suppositions  the  number  of  stars  posited  at  * 

*  See  Silliman's  Journal,  vol.  xlv.  No.  I,  page  206. 
t  Herschel's  Outlines  of  Astronomy,  p.  520. 


212 


OF    THE    FIXED    STARS. 


any  given  distance,  and  having  therefore  the  same  apparent  mag- 
nitude, will  be  greater  in  proportion  as  this  distance  is  greater,  and 
thus  as  the  apparent  magnitude  is  lower. 

565.  It  is  not  to  be  understood  that  the  classification  of  the  stars 
into  different  magnitudes  is  made  according  to  any  fixed  definite 
proportion  subsisting  between  the  degrees  of  apparent  brightness 
of  the  stars  belonging  to  different  classes.     Stars  of  almost  every 
gradation  of  brightness,  between  the  highest  and  the  lowest,  are 
met  with.     Those  which  offer  marked  differences  of  lustre,  form 
the  basis  of  the  classification  ;  others,  which  do  not  differ  very 
widely  from  these,  are  united  to  them.     As  a  necessary  conse- 
quence, there  are  some  stars  of  intermediate  lustre,  which  cannot 
be  assigned  with  certainty  to  either  magnitude.    Thus,  in  the  cata- 
logue of  the  Astronomical  Society  of  London,  3  stars  are  marked 
as  intermediate  between  the  first  and  second  magnitudes,  and  29 
betw.een  the  second  and  third. 

Different  astronomers  also  not  unfrequently  assign  the  same  star 
to  different  magnitudes. 

As  to  the  proportions  of  light  emitted  from  the  average  stars  of 
the  different  magnitudes,  according  to  the  experimental  comparisons 
of  Sir  Wm.  Herschel,  they  are,  from  the  first  to  the  sixth  magni- 
tude, approximately  in  the  ratio  of  the  numbers,  100,  25,  12, 6, 2, 1 . 

566.  With  the  exception  of  the  three  or  four  brightest  classes, 
the  stars  are  not  distributed  indiscriminately  over  the  sphere  of  the 
heavens,  but  are  accumulated  in  far  greater  numbers  on  the  borders 
of  that  belt  of  cloudy  light  in  the  heavens,  which  is  called  the 
milky  way,  and  in  the  milky  wray  itself,  which  the  telescope  shows 
to  consist  of  an  immense  number  of  stars  of  small  magnitude  in 
close  proximity. 

Herschel  found  that  on  a  medium  estimate  a  segment  of  the 
milky  way,  15°  long,  and  2°  broad,  contained  at  least  50,000  stars 
of  sufficient  magnitude  to  be  distinguished  through  his  telescope.* 
According  to  this,  taking  its  average  breadth  at  14°,  the  milky  way 
must  contain  more  than  eight  millions  of  stars. 

567.  This  great  accumulation  of  stars  in  a  zone  of  the  heavens, 
encompassing  the  earth  in  the  direction  of  a  great  circle,  suggested 
to  the  mind  of  Herschel  the  idea  that  the  stars  of  our  firmament 
are  not  disseminated  indifferently  throughout  the  surrounding  re- 
gions of  space,  but  are  for  the  most  part  arranged  in  a  stratum, 
the  thickness  of  which  is  very  small  in  comparison  with  its  breadth  ; 


Fig.  100. 


— the  sun  and  solar  sys- 
tem being  near  the  mid- 
dle of  the  thickness.  If 
S  (Fig.  100)  represents 
the  place  of  the  sun,  it 
will  be  seen  that  upon 
this  supposition  the 


*  A  Newtonian  reflecting  telescope  of  20  feet  focus  and  nearly  19  in.  in  aperture. 


ANNUAL  PARALLAX  OF  THE  STARS, 


213 


number  of  stars  in  the  direction  SC  of  the  thickness  of  the  stratum 
will  be  less  than  in  any  other  direction,  and  that  the  greatest  num- 
ber will  lie  in  the  direction  of  the  breadth,  as  SB.  On  one  side 
of  the  point  S,  the  stratum  is  supposed  to  be  divided  for  a  cer- 
tain distance  into  two  laminae,  as  Fig.  101. 
shown  in  the  figure,  which  repre- 
sents a  section  of  the  supposed  stra- 
tum. This  supposition  is  necessary 
to  account  for  the  two  branches, 
with  a  dark  space  between  them, 
into  which  the  milky  way  is  divided 
for  about  one-third  of  its  course. 

Herschel  undertook  to  gauge  this  stratum 
in  various  directions,  on  the  principle  that 
the  distance  through  to  its  borders  in  any 
direction  was  greater  in  proportion  as  the 
number  of  stars  seen  in  that  direction  was 
greater.  He  thus  found  that  its  actual  form 
was  very  irregular :  its  section,  instead  of 
being  truly  that  of  a  segment  of  a  sphere 
divided  for  a  certain  distance  into  two  lami- 
nae, as  represented  in  Fig.  100,  having  the 
form  represented  in  Fig.  101.  He  estimated 
the  thickness  of  the  stratum  to  be  less  than 
160  times  the  interval  between  the  stars,  and 
the  breadth  to  be  nowhere  greater  than  1000 
times  the  same  distance.  He  conceived  that 
it  extended  in  no  direction  a  distance  equal 
to  tiie  space-penetrating  power  of  his  tele- 
scope for  individual  stars,  and  much  less  for 
collections  of  stars  seen  as  nebulous  spots. 

568.  Sir  John  Herschel  conceives 
that  the  superior  brilliancy  and 
larger  development  of  the  milky 
way  in  the  southern  hemisphere, 
from  the  constellation  Orion  to  that 
of  Antinous,  indicate  that  the  sun 
and  his  system  are -at  a  distance  from 
the  centre  of  the  stratum  in  the  di- 
rection of  the  Southern  Cross,  and 
that  the  central  parts  are  so  vacant 
of  stars  that  the  whole  approximates 
to  the  form  of  an  annulus. 


ANNUAL  PARALLAX  AND  DISTANCE  OF  THE  STARS. 

v 

569.  The  Annual  Parallax  of  a  fixed  star  is  the  angle  made  by 
two  lines  conceived  to  be  drawn,  the  one  from  the  sun  and  the 
other  from  the  earth,  and  meeting  at  the  star,  at  the  time  the  earth 
is  in  such  part  of  its  orbit  that  its  radius-vector  is  perpendicular  to 
the  latter  line  ;  or,  in  other  words,  it  is  the  greatest  angle  that  can 


214  OF   THE    FIXED    STARS. 

be  subtended  at  the  star  by  the  radius  of  the  earth's  orbit.  Thus, 
let  S  (Fig.  102)  be  the  sun,  s  a  fixed  star,  and  E  the  earth,  in 
such  a  position  that  the  radius-vector  SE  is  perpendicular  to  Es 
Fig.  102.  _  the  line  of  direction  of  the  star, 

then  the  angle  SsE  is  the  an- 
nual parallax  of  the  star  s. 

570.  If  the  annual  parallax 
of  a  star  was  known,  we  might 
easily  find  its  distance  from  the 
earth ;  for  in  the  right-angled  tri- 
angle SEs  we  would  know  the 
angle  SsE  and  the  side  SE,  and 
we  should  only  have  to  com- 
pute the  side  Es.  Now,  if  any 
of  the  fixed  stars  have  a  sensi- 
ble parallax,  it  could  be  detected  by  a  comparison  of  the  places  of 
the  star,  as  observed  from  two  positions  of  the  earth  in  its'  orbit, 
diametrically  opposite  to  each  other ;  and  accordingly,  the  atten- 
tion of  astronomers  furnished  with  the  most  perfect  instruments, 
has  long  been  directed  to  such  observations  upon  the  places  of 
some  of  the  fixed  stars,  in  order  to  determine  their  annual  paral- 
lax. But,  after  exhausting  every  refinement  of  observation,  they 
have  not  been  able  to  establish  that  any  of  them  have  a  measura- 
ble parallax.  Now,  such  is  the  nicety  to  which  the  observations 
have  been  carried,  that,  did  the  angle  in  question  amount  to^  as 
much  as  1",  it  could  not  possibly  have  escaped  detection  and  uni- 
versal recognition.  We  may  then  conclude  that  the  annual  par- 
allax of  the  nearest  fixed  star  is  less  than  \" . 

571.  Taking  the  parallax  at  1",  the  distance  of  the  star  comes 
out  206,265  times  the  distance  of  the  sun  from  the  earth,  or  about 
20  billions  of  miles.     The  distance  of  the  nearest  fixed  star  must 
therefore  be  greater  than  this.     A  juster  notion  of  the  immense 
distance  of  the  fixed. stars,  than  can  be  conveyed  by  figures,  may 
be  gained  from  the.  consideration  that  light,  which  traverses  the 
distance  between  the  sun  and  earth  in  8m.  18s.,  and  would  perform 
the  circuit  of  our  globe  in  }  of  a  second,  employs  more  than  three 
years  in  coming  from  the  nearest  fixed  star  to  the  earth. 

According  to  Struve,  the  most  probable  value  of  the  parallax  of  a  star  of  the 
first  magnitude  is  no  more  than  about  |"  ;  which  would  make  its  distance  5  times 
greater  than  the  above  determination. 

572.  The  statement  made  in  Art.  570,  that  the  annual  parallax  of  the  fixed  stars 
has  hitherto  escaped  certain  detection,  although  truly  representing  the  result  of  all 
the  many  efforts  made  to  solve  the  great  problem  of  the  distance  of  the  fixed  stars, 
until  a  very  recent  date,  is  at  the  present  time  (1845)  no  longer  true.     The  paral- 
lax of  one  of  the  fixed  stars  is  now  believed  to  have  been  determined  by  Bessel. 
This  is  the  star  61  Cygni.*     It  is  a  star  of  about  the  6th  magnitude,  barely  visible 
to  the  naked  eye.     When  viewed  through  a  telescope  it  is  seen  to  consist  of  two 
stars  of  nearly  equal  brightness,  at  a  distance  from  each  other  of  about  16".  These 

*  *  R.  A.  314°  52',  Dec.  N.  37°  56'. 


DISTANCE  OF  THE  STARS.  215 

stars  have  a  motion  of  revolution  around  each  other,  and  the  two  move  together 
at  the  same  rate,  of  5".3  per  year,  as  one  star,  along  the  sphere  of  the  heavens.  It 
is  hence  inferred  that  they  are  bound  together  into  one  system  by  the  principle  of 
gravitation,  and  are  at  pretty  nearly  the  same  distance  from  the  earth.  The  great 
proper  motion  of  this  double  star,  as  compared  with  other  stars,  led  to  the  suspicion 
that  it  was  nearer  than  any  other ;  and  thus  to  attempts  to  determine  its  parallax. 
The  principle  of  Bessel's  method  is  to  find  the  difference  between  the  parallaxes  of 
the  star  61  Cygni,  and  some  other  star  of  much  smaller  magnitude,  and  therefore 
supposed  to  be  at  a  much  greater  distance,  seen  in  as  nearly  the  same  direction  as 
possible.  -This  difference  will  differ  from  the  absolute  parallax  of  the  double  star 
by  only  a  small  fraction  of  its  whole  amount.  It  was  found  by  measuring  with  a 
position  micrometer  (76)  the  annual  changes  in  the  distance  of  the  two  stars,  and 
in  the  position  of  the  line  joining  them.  To  make  it  evident  that  such  changes 
will  be  an  inevitable  consequence  of  any  difference  of  parallax  in  the  two  stars, 
conceive  two  cones  having  the  earth's  orbit  for  a  common  base,  and  their  vertices 
respectively  at  the  two  stars,  and  imagine  their  surfaces  to  be  produced  past  the 
stars  until  they  intersect  the  heavens.  The  intersections  will  be  ellipses,  but,  by 
reason  of  the  different  distances  of  the  two 
stars,  of  different  sizes,  as  represented  in  Fig. 
103  ;  and  they  will  be  apparently  described  by 
the  stars  in  the  course  of  one  revolution  of  the 
earth  in  its  orbit.  The  two  stars  will  always 
be  similarly  situated  in  their  parallactic  ellip- 
ses :  thus,  if  one  is  at  A  the  other  will  be  at 
a  ;  and  after  the  earth  has  made  one-quarter 
of  a  revolution,  they  will  be  at  B  and  b  ;  and  af- 
ter another  quarter  of  a  revolution  at  C  and  c, 
&c.  Now  it  will  be  manifest,  on  inspecting 
the  figure,  the  ellipses  being  of  unequal  size, 
that  the  line  of  the  stars  will  be  of  unequal 
lengths,  and  have  different  directions  in  the 
different  situations  of  the  stars. 

A  much  smaller  angle  of  parallax  may  be 
found,  with  the  same  degree  of  certainty,  by 
this  indirect  method  than  by  the  direct  process 
explained  in  Art.  570 ;  for,  since  the  two  stars  are  seen  in  pretty  nearly  the  same 
direction,  they  will  be  equally  affected  by  refraction  and  aberration  ;  and  since  it 
is  only  the  relative  situations  of  the  two  stars  that  are  measured,  no  allowance  haa 
to  be  made  for  precession  and  nutation,  or  for  errors  in  the  construction  or  adjust- 
ment of  the  instrument.  It  is  therefore  independent  of  the  errors  that  are  inevita- 
bly committed  in  the  determination  of  these  several  corrections,  when  it  is  at- 
tempted to  find  directly  the  absolute  parallax,  by  observing  the  right  ascension  and 
declination  at  opposite  seasons  of  the  year.  The  measurements  made  with  the 
micrometer  in  the  hands  of  the  most  accurate  observers,  may  be  relied  on  as  exact 
to  within  a  small  fraction  of  1". 

For  the  sake  of  greater  certainty  Bessel  made  the  measurements  of  parallactic 
changes  of  relative  situation  between  the  star  61  Cygni,  and  two  small  stars  in- 
stead of  one, — the  middle  point  between  the  two  members  of  the  double  star  being 
taken  for  the  situation  of  this  star.  He  found  the  difference  of  parallax  to  be  for 
the  one  star  0".3584,  and  for  the  other  star  0".3289  :  and  assuming  the  absolute 
parallax  of  the  two  stars  to  be  equal,  found  for  the  most  probable  value  of  the  dif- 
ference of  parallax  0".3483.  Whence  he  calculated  the  distance  of  the  star  61 
Cygni  to  be  592,200  times  the  mean  distance  of  the  earth  from  the  sun  ;  a  distance 
which  would  be  traversed  by  light  in  9£  years.  (See  Note  XIV.) 

573.  The  amount  of  light  received  from  the  same  body  at  differ- 
ent distances  varies  inversely  as  the  square  of  the  distance.  Hence, 
if  we  admit  the  light  of  a  star  of  each  magnitude  to  be  half  that  of 
one  of  the  next  higher  magnitude,  a  star  of  the  first  magnitude 
would  have  to  be  removed  to  360  times  its  distance,  to  appear  no 
brighter  than  one  of  the  eighteenth.  Accordingly,  if  the  difference 


216  OF    THE    FIXED    STARS. 

in  the  apparent  magnitude  of  the  stars  arises  for  the  most  part 
from  a  difference  of  distance,  (which  is  the  more  probable  suppo- 
sition,) there  must  be  a  multitude  of  stars  visible  in  telescopes, 
the  light  of  which  has  taken  at  least  one  thousand  years  to  reach 
the  earth. 

A  calculation  based  upon  the  power  of  large  telescopes  to  aug- 
ment the  amount  of  light  received  from  the  stars,  in  connection  with 
the  well-known  law  of  diminution  of  the  light  received  as  the  dis- 
tance increases,  conducts  to  about  the  same  result. 

'NATURE  AND  MAGNITUDE  OF  THE  STARS. 

574.  The  vast  distance  at  which  the  fixed  stars  are  visible,  and 
shine  with  a  light  not  much  inferior  to  the  planets,  leaves  no  room 
to  doubt  that  they  are  all  suns  like  our  own.     If  it  should  be  con- 
jectured that  some  of  the  fainter  stars  might  be  bodies   shining 
by  reflected  light,  like  the  planets,  the  answer  is,  that  if  we  were 
to  suppose  the  existence  ef  opake  bodies,  at  the  distance  of  the 
stars,  so  inconceivably  vast  in  their  dimensions  as  to  send  a  sensi- 
ble light  to  the  eye,  if  illuminated  to  the  same  degree  as  the  plan- 
ets, the  stars  of  the  smaller  magnitudes  are,  with  the  exception 
perhaps  of  the  members  of  some  of  the  double  stars,  too  remote 
from  the  brighter  ones  to  receive  sufficient  light  from  them ;  for, 
the  smallest  measurable  space  in  the  field  of  the  largest  telescopes 
is,  at  the  distance  of  the  nearest  star,  as  large  as  or  larger  than  the 
earth's   orbit.     It  is  perhaps  possible,  that  some  of  the  faintest 
members  of  some  of  the  double  stars,  as  surmised  by  Sir  John 
Herschel,  may  shine  by  reflected  light.    '  ?•• 

575.  To  be  able  to  determine  the  magnitude  of  a  star,  we  must 
know  its  distance,  and  also  its  apparent  diameter.     Now  the  dis- 
tance of  only  one  star  has,  as  yet,  been  found ;  and  the  discs  of 
all  the  stars,  even  in  the  most  powerful  telescopes,  are  altogether 
spurious  ;  so  that  in  no  instance  have  we  the  data,  nor  have  we 
reason  to  expect  that  they  will  be  hereafter  obtained,  for  determin- 
ing with  certainty  the  magnitude  of  a  fixed  star. 

We  may  infer,  however,  from  the  intensity  of  their  light,  that  it 
is  highly  probable  that  some  at  least  of  the  stars  are  as  large  as, 
or  even  larger  than  the  sun.  It  has  been  calculated  from  the  re- 
sults of  photometrical  experiments  made  by  Dr.  Wollaston,  on  the 
relative  quantity  of  light  received  from  Sirius  and  the  sun,  that  if 
the  sun  were  removed  to  the  distance  of  20  billions  of  miles,  which 
is  known  to  be  less  than  the  distance  of  any  of  the  stars,  he  would 
not  send  to  us  so  much  as  half  the  quantity  of  light  actually  re- 
ceived from  Sirius. 

576.  Although  there  are  not  sufficient  data  for  calculating  the  magnitude  of  the 
star  61  Cygni,  there  are  for  ascertaining  its  mass.  This  element  results  from  the 
distance  and  the  motion  of  revolution  of  the  two  members  of  the  double  star  about 
each  other.  Bessel  finds  it  to  be  less  than  half  of  the  sun's  mass.  According  to 


VARIABLE    STARS.  217 

this  result  the  sun,  as  seen  from  this  star,  should  appear  as  a  star  of  about  the  fifth 
magnitude. 

'  *      4 

VARIABLE   STARS. 

577.  A  number  of  the  fixed  stars  are   subject  to  periodical 
changes  of  brightness,  and  are  hence  called  Variable  Stars,  or 
Periodical  Stars.     One  of  the  most  remarkable  of  the  variable 
stars  is  the  star  Omicron,  in  the  constellation  Cetus.     From  being 
as  bright  as  a  star  of  the  second  magnitude,  it  gradually  decreases 
until  it  entirely  disappears  ;  and,  after  remaining  for  a  time  invisi- 
ble, reappears,  and  gradually  increasing  in  lustre,  finally  recovers 
its  original  appearance.    The  period  of  these  changes  is  332  days. 
It  remains  at  its  greatest  brightness  about  two  weeks,  employs 
about  three  months  in  waning  to  its  disappearance,  continues  invi- 
sible for  about  five  months,  and  during  the  remaining  three  months 
of  its  period  increases  to  its  original  lustre.     Such  is  the  general 
course  of  its  phases..    It  does  not,  however,  always  recover  the 
same  degree  of  brightness,  nor  increase  and  dimmish  by  the  same 
gradations.     It  is  related  by  Hevelius,  that  in  one  instance  it  re- 
mained invisible  for  a  period  of  four  years,  viz.  from  October,  1672, 
to  December,  1676.*     A  similar  phenomenon  has  been  noticed  in 
the  case  of  another  variable  star,  viz.  the  star  •%  Cygni.     It  is  sta- 
ted by  Cassini  to  have  been  scarcely  visible  throughout  the  years 
1699,  1700,  and  1701,  at  those  times  when  it  ought  to  have  been 
most  conspicuous.     On  the  other  hand,  a  variable  star,  situated  in 
the  Northern  Crown,  sometimes  continues  visible  for  several  years 
without  any  apparent  change,  and  then  resumes  its  regular  varia- 
tions. 

578.  The  greater  number  of  variable  stars  undergo  a  regular 
increase  and  diminution  of  lustre,  without  ever,  like  the  star  just 
noticed,  becoming  entirely  invisible.  The  star  Algol,  or  ft  Perseii, 
is  a  remarkable  variable  star  of  this  description.     For  a  period  of 
2d.  14h.  it  appears  as  a  star  of  the  second  magnitude,  after  which 
it  suddenly  begins  to  diminish  in  splendor,  and  in  about  3|-  hours 
is  reduced  to  a  star  of  the  fourth  magnitude.    It  then  begins  again 
to  increase,  and  in  3|  hours  more  is  restored  to  its  usual  bright- 
ness, going  through  all  its  changes  in  2d.  20h.  48m.f 

579.  There  are  also  a  number  of  double  stars,  one  or  both  of 
the  members    of  which   are   variable ;  as  y  Virginis,  s   Arietis, 
£  Bootis,  &c. 

580.  Two  general  facts  have  been  noticed  with  respect  to  the 
variable  stars,  which  are  worthy  of  remark,  viz.  that  the  color  of 
their  light  is  red,  and  that  their  phase  of  least  light  lasts  much  long 
er  than  that  of  their  greatest  light.  The  star  Algol,  which  is  white, 
is  said  to  be  the  only  variable  star  whose  light  is  not  of  a  reddish 

*  Herschel's  Treatise  on  Astronomy,  p.  356.         t  Ibid.  357. 

28 


218  OP  THE  FIXED  STARS. 

color.    The  same  star  also  presents  an  exception  to  the  other  gen- 
eral fact  just  noticed.     (See  Note  XV.) 

581.  There   are  also  some  instances  on  record  of  temporary 
stars  having  made  their  appearance  in  the  heavens  ;  breaking  forth 
suddenly  in  great  splendor,  and  without  changing  their  positions 
among  the  other  stars,  after  a  time  entirely  disappearing.    One  of 
the  most  noted  of  these  is  the  star  which  suddenly  shone  forth  with 
great  brilliancy  on  the  llth  of  November,  1572,  between  the  con- 
stellations Cepheus  and  Cassiopeia,  and  was  attentively  observed 
by  Tycho  Brahe.     It  was  then  as  bright  as  any  of  the  permanent 
stars,  and  continued  to  increase  in  splendor  till  it  surpassed  Jupiter 
when  brightest,  and  was  visible  at  mid-day.     It  began  to  diminish 
in  December  of  the  same  year,  and  in  March,  1 574,  it  entirely  dis- 
appeared, after  having  remained  visible  for  sixteen  months,  and 
has  not  since  been  seen.* 

It  was  noticed  that  while  visible  the  color  of  its  light  changed  from  white  to  yel- 
low, and  then  to  a  very  distinct  red ;  after  which  it  became  pale,  like  Saturn. 

In  the  years  945  and  1264,  brilliant  stars  appeared  in  the  same 
region  of  the  heavens.  It  is  conjectured  from  the  tolerably  near 
agreement  of  the  intervals  of  the  appearance  of  these  stars  and 
that  of  1572,  that  the  three  may  be  one  and  the  same  star,  with  a 
period  of  about  300  years.  The  places  of  the  stars  of  945  and 
1264  are,  however,  too  imperfectly  known  to  establish  this  with 
any  degree  of  certainty. 

Besides  these  three  temporary  stars,  several  others  have  made  their  appearance, 
viz.  one  in  the  year  125  B.  C.,  seen  by  Hipparchus ;  another  in  389  A.  D.,  in  the 
constellation  Aquila  ;  a  third  in  the  9th  century,  in  Scorpio;  a  fourth  in  1604,  in 
Serpentarius,  seen  by  Kepler;  and  a  fifth  in  1670,  in  the  Swan. 

582.  What  is  no  less  remarkable  than  the  changes  we  have  no- 
ticed, several  stars,  which  are  mentioned  by  the  ancient  astrono- 
mers, have  now  ceased  to  be  visible,  and  some  are  now  visible 
to  the  naked  eye  which  are  not  in  the  ancient  catalogues. 

583.  The  most  probable  explanation  of  the  phenomenon  of  variable  stars,  is,  that 
they  are  self-luminous  bodi<  s  rotating  upon  axes,  like  the  sun,  and  having  like  him 
spots  upon  their  surface,  bot  vastly  larger  and  more  permanent.     By  the  rotation 
these  spots  are  brought  periodically  around  on  the  side  towards  the  earth,  and  ac- 
cording to  their  size  occasion  a  diminution  of  the  light  of  the  star,  or  make  it  en- 
tirely to  disappear.    In  the  case  of  the  star  Algol,  however,  as  suggested  by  Good- 
ricke,  the  phenomena  are  precisely  such  as  would  result  from  the  periodical  inter- 
position of  an  opake  body  revolving  around  it.     In  those  cases  in  which  the  period 
of  the  diminution  of  the  light  is  a  large  fraction  of  the   entire  period  of  the  star, 
(580,)  as  well  as  those  in  which  there  are  occasional  interruptions  in  the  regular 
recurrence  of  the  phenomena,  (577,)  the  supposition  of  the  interposition  of  an  opake 
body  will  not  answer.    (See  Note  XVL) 

584.  Temporary  stars  are  most  probably  suns  which  have  entirely  intermitted 
the  evolution    of  light  for  a  long  period  of  time,  and  then  burst  forth  anew  with  a 
sudden  and  peculiar  splendor.     Laplace  conjectured  that  they  might  be  the  confla- 
grations of  distant  worlds  ;  but  it  seems  very  questionable  whether  the  conflagra- 

*  Herschel's  Treatise  on  Astronomy,  p.  359. 


DOUBLE   STARS.  219 

tion  of  even  an  entire  system  of  planets  would  furnish  as  much  light  as  the  sun  at 
its  centre ;  and  the  large  and  permanent  spots  on  the  surface  of  the  variable  stars 
would  seem  to  render  it  probable  that  some  suns  have  become,  for  a  time,  entirely 
extinct.  In  support  of  this  theory,  that  temporary  stars  are  the  temporary  revival 
of  extinct  suns,  we  have  the  fact  said  to  have  been  recently  discovered  by  Bessel, 
that  there  are  opake  bodies  in  space  of  the  size  of  suns.  It  is  stated  that  this 
distinguished  astronomer  has  ascertained,  from  a  discussion  of  the  most  accurate 
observations  that  have  been  made  upon  these  stars,  that  the  proper  motions  of  the 
two  stars  Sirius  and  Procyon  deviate  sensibly  from  uniformity ;  whence  he  infers 
that  they  must  each  be  revolving  about  some  large  non-luminous  body  in  their  vi- 
cinity, and  are  thus  double  stars,  one  of  the  members  of  which  is  non-luminoue. 

DOUBLE  STARS. 

585.  Many  of  the  stars  which  to  the  naked  eye  appear  single, 
when  examined  with  telescopes  are  found  to  consist  of  two  (in 
some  instances  three  or  more)  stars  in  close  proximity  to  each  oth- 
er. These  are  called  Double  Stars,  or  Multiple  Stars.  (See  Fig. 
104.)  This  class  of  bodies  was  first  attentively  observed  by  Sir 
William  Herschel,  who,  in  the  years  1782  and  1785,  published 

Fig.  104. 


Castor.        y  Leonis.        Rigel.  Pole-star.       11  Monoc.      $  Cancri. 

catalogues  of  a  large  number  of  them  which  he  had  observed.  The 
list  has  since  been  greatly  increased  by  Professor  Struve,  of  Dor- 
pat,  Sir  J.  F.  W.  Herschel,  and  other  observers,  and  now  amounts 
to  several  thousand. 

586.  Double  stars  are  of  various  degrees  of  proximity.  In  a  great 
number  of  instances,  the  angular  distance  of  the  individual  stars  is 
less  than  1'',  and  the  twojcan  only  be  separated  by  the  most  pow- 
erful telescopes.     In  other  instances,  the  distance  is  |'  and  more, 
and  the  separation  can  be  effected  with  telescopes  of  very  moder- 
ate power.     They  are  divided  into  different  classes  or  orders,  ac- 
cording to  their  distances ;  those  in  which  the  proximity  is  the 
closest  forming  the  first  class. 

587.  The  two  members  of  a  double  star  are  generally  of  quite 
unequal  size.  (See  Fig.  104.)     But  in  some  instances,  as  that  of 
the  star  Castor,  they  are  of  nearly  the  same  magnitude.     Double 
stars  occur  of  every  variety  of  magnitude. 

588.  It  is  a  curious  fact,  that  the  two  constituents  of  a  double  star  in  numer- 
ous instances  shine  with  different  colors  ;  and  it  is  still  more  curious  that  these 
colors  are  in  general    complementary  to  each  other.      Thus,  the  larger   star  is 
usually  of  a  ruddy  or  orange  hue,  while  the  smaller  one   appears  blue  or  green. 
This  phenomenon  has  been  supposed  to  be  in  some  cases  the  effect  of  contrast ; 
the  larger  star  inducing  the  accidental  color  in  the  feebler  light  of  the  other.     "Sir 
John  Herschel  cites  as  probable  examples  of  this  effect  the  two  stars  t  Cancri,  and 
yAndromedoe.      But  it  is  maintained  by  Nichol  that  this  explanation  cannot  be 
admitted  ;  for,  if  true,  it  ought  to  be  universal,  whereas  there  are  many  systems 
similar  in  relative  magnitudes  to  the  contrasted  ones,  in  which  both  stars  are  yel- 
low, or  otherwise  belong  to  the  red  end  of  the  spectrum.     Again,  if  the  blue  01 


220  OF  THE  FIXED  STARS. 

violet  color  were  the  effect  of  contrast,  it  ought  to  disappear  when  the  yellow  star 
is  hid  from  the  eye ;  which,  however,  it  does  not  do.  Thus,  the  star  /8  Cygni  con- 
sists of  two  stars,  of  which  one  is  yellow,  and  the  other  shines  with  an  intensely 
blue  light ;  and  when  one  of  them  is  concealed  from  view  by  an  interposed  slip  of 
darkened  copper,  the  other  preserves  its  color  unchanged.  The  color,  then,  of 
neither  of  the  stars  can  be  accidental. 

It  may  be  remarked  in  this  connection,  that  the  isolated  stars  also  shine  with 
various  colors.  For  example,  among  stars  of  the  first  magnitude,  Sirius,  Vega, 
Altair,  Spica  are  white,  Aldebaran,  Arcturus,  Betelgeux  red,  Capella  and  Procy- 
on  yellow.  In  smaller  stars  the  same  difference  is  seen,  and  with  equal  distinct- 
ness when  they  are  viewed  through  telescopes.  According  to  Herschel,  insulated 
stars  of  a  red  color,  almost  as  deep  as  that -of  blood,  occur  in  many  parts  of  the 
heavens,  but  no  decidedly  green  or  blue  star  has  ever  been  noticed  unassociatcd 
with  a  companion  brighter  than  itself. 

589.  Sir  William  Herschel  instituted  a  series  of  observations 
upon  several  of  the  double    stars,  with  the  view  of  ascertaining 
whether  the  apparent  relative  situation  of  the  individual  stars  expe- 
rienced any  change  in  consequence  of  the  annual  variation  of  the 
parallax  of  the  star.     With  a  micrometer  adapted  to  the  purpose, 
(76,)  he  measured  from  time  to  time  the  apparent  distance  of  the 
two  stars,  and  the  angle  formed  by  their  line  of  junction  with  the 
meridian  at  the  time  of  the  meridian  passage,  called  the  Angle  of 
Position.     Instead,  however,  of  finding  that  annual  variation  of 
these  angles,  which  the  parallax  of  the  earth's  annual  motion  would 
produce,  he  observed  that,  in  many  instances,  they  were  subject  to 
regular  progressive  changes,  which  seemed  to  indicate  a  real  mo- 
tion of  the  stars  with  respect  to  each  other.  After  continuing  his  ob- 
servations for  a  period  of  twenty-five  years,  he  satisfactorily  ascer- 
tained that  the  changes  in  question  were  in  reality  produced  by  a 
motion  of  revolution  of  one  star  around  the  other,  or  of  both  around 
their  common  centre  of  gravity ;  and  in  two  papers,  published  in 
the  Philosophical  Transactions  for  the  years   1803  and  1804,  he 
announced  the  important  discovery  that*  there  exist  sidereal  sys- 
tems composed  of  two  stars  revolving  about  each  other  in  regular 
orbits.    These  stars  have  received  the  appellation  of  Binary  Stars t 
to  distinguish  them  from  other  double  stars  which  are  not  thus 
physically  connected,  and  whose  apparent  proximity  may  be  occa- 
sioned by  the  circumstance  of  their  being  situated  on  nearly  the 
same  line  of  direction  from  the  earth,  though  at  very  different  dis- 
tances from  it.     Similar  stars,  consisting  of  more  than  two  consti- 
tuents, are  called  Ternary,  Quaternary,  &c. 

590.  Since  the  time  of  Sir  W.  Herschel,  the  observations  upon 
the  binary  stars  have  been  continued  by  Sir  John  Herschel,  Sir 
James  South,  Struve,  Bessel,  Madler,  and  other  astronomers  :  ac- 
cording to  Madler,  the  number  of  known  binary  and  ternary  stars 
is  now  about  250.     Every  year  materially  increases  the  list ;  and 
will  probably  continue  to  do  so  for  some  time  to  come  :  for,  while 
the  changes  of  relative  situation  are  in  some  instances  exceedingly 
slow,  the  actual  number  of  such  systems  is  probably  a  large  frac- 
tion of  the  whole  number  of  double  stars ;  at  least,  if  we  confine 


DOUBLE  STARS.  221 

our  attention  to  double  stars  whose  constituents  are  within  y  of 
each  other.  This  may  be  inferred  from  the  fact,  that  the  num- 
ber of  such  double  and  multiple  stars  actually  observed,  which 
amounts  to  over  3000,  is  at  least  ten  times  greater  than  the  num- 
ber of  instances  of  fortuitous  juxtaposition  that  would  obtain  on  the 
supposition  of  a  uniform  distribution  of  the  stars.  Besides,  there 
is  a  number  of  double  stars  not  yet  discovered  to  have  a  motion  of 
revolution,  which  still  give  indications  of  a  physical  connection. 
Thus,  their  constituents  are  found  to  have  constantly  the  same 
proper  motion  in  the  same  direction  ;  showing  that  they  are  in  all 
probability  moving  as  one  system  through  space. 

Prom  the  observations  made  upon  some  of  the  binary  stars,  as- 
tronomers have  been  enabled  to  deduce  the  form  of  their  orbits, 
and  approximately  the  lengths  of  their  periods.  The  orbits  arc 
ellipses  of  considerable  eccentricity.  The  periods  are  of  various 
lengths,  as  will  be  seen  from  the  following  enumeration  of  those 
which  are  considered  as  the  best  ascertained  :  tf  Coronas  608  years  ; 
61  Cygni  540  years ;  a  Geminorum  232  years ;  7  Virginis  182 
years  ;  3062  Stmve  95  years  ;  p  Ophiuchi  93  years  ;  X  Ophiuchi 
88  years  ;  w  Leonis  83  years  ;  g  Ursae  Majoris  60  years  ;  £  Can- 
cri  59  years  ;  n  Coronae  43  years  ;  £  Herculis  31  years.  Fig.  105 
represents  a  portion  of  the  apparent  orbit  of  the  double  star  y  Vir- 
ginis, and  shows  the  relative  positions  of  the  two  members  of  the 

Fig.  105. 


double  star  in  various  years.  At  the  time  of  their  nearest  approach, 
in  1836,  the  interval  between  them  was  a  fraction  of  1",  and  they 
could  not  be  separated  by  the  best  telescopes,  with  a  magnifying 
power  of  1000.  Since  then  their  distance  has  been  continually 
increasing.  In  1844  it  amounted  to  2",  and  a  power  of  from  200 
to  300  was  sufficient  to  separate  them.  The  orbit  represented  in 
the  figure  is  the  stereographic  projection  of  the  true  orbit  on  a 
plane  perpendicular  to  the  line  of  sight.  (See  Note  XVII.) 


222  OP  THE  FIXED  STARS. 

The  actual  distance  between  the  members  of  a  binary  star  hag 
been  found  only  for  the  star  61  Cygni.  Bessel  makes  it  for  this 
star  about  two  and  a  half  times  the  distance  of  Uranus  from  the  sun. 

591.  It  is  important  to  observe,  that  the  revolution  of  one  star 
around  another  is  a  different  phenomenon  from  the  revolution  of  a 
planet  around  the  sun.     It  is  the  revolution  of  one  sun  around  an- 
other sun ;  of  one  solar  system  around  another  solar  system ;  or 
rather  of  both  around  their  common  centre  of  gravity.    We  learn 
from  it  the  important  fact,  that  the  fixed  stars  are  endued  with  the 
same  property  of  attraction  that  belongs  to  the  sun  and  planets. 

PROPER  MOTIONS  OF  THE  STARS. 

592.  It  has  already  been  stated  (181)  that  the  fixed  stars,  so 
called,  are  not  all  of  them  rigorously  stationary.     By  a  careful 
comparison  of  their  places,  found  at  different  times  with  the  accu- 
rate instruments  and  refined  processes  of  modern  observation,  it 
has  been  found  that  great  numbers  of  them  have  a  progressive  mo- 
tion along  the  sphere  of  the  heavens,  from  year  to  year.     The  ve- 
locity and  direction  of  this  motion  are  uniformly  the  same  for  the 
same  star,  but  different  for  different  stars.     The  star  which  has 
the  greatest  proper  motion  of  any  observed,  is  the  double  star  61 
Cygni.    During  the  last  fifty  years  it  has  shifted  its  position  in  the 
heavens  4'  23"  ;  the  annual  proper  motion  of  each  of  the  individ- 
ual stars  being  5". 3.    Among  isolated  stars,  j^  Cassiopeiae  has  the 
greatest  proper  motion.     It  changes  its  place  3". 74  every  year. 
The  proper  motions  of  some  of  the  stars  are  either  partially  or  en- 
tirely attributable  to  a  motion  of  the  sun  and  the  whole  solar  sys- 
tem in  space  ;  but  the  motions  of  others  cannot  be  reconciled  with 
this  hypothesis,  and  must  be  regarded  as  in  all  probability  indica- 
tive of  a  real  motion  of  these  bodies  in  space.  (See  Note  XVIII.) 

593.  The  first  successful  attempt  to  explain  the  proper  motions 
of  the  fixed  stars  on  the  hypothesis  of  a  motion  of  the  solar  system 
through  space,  was  made  by  Sir  William  Herschel.    After  a  care- 
ful examination  of  these  motions,  he  conceived  that  the  majority 
of  them  could  be  explained  on  the  supposition  of  a  general  recess 
of  the  stars  from  a  point  near  that  occupied  by  the  star  X  Herculis 
towards  a  point  diametrically  opposite.     Whence  he  inferred  that 
the  sun  with  its  attendant  system  of  planets  was  moving  rapidly 
through  space  in  a  direction  towards  this  constellation.    Doubt  has 
since  been  thrown  upon  these  conclusions  by  Bessel  and  other  as 
tronomers  ;  but  they  have  quite  recently  been  decisively  re-estab- 
lished by  M.  Argelander,  of  Abo.     The  investigations  of  Arge- 
lander,  which  were  communicated  to  the  Academy  of  St.  Peters- 
burgh  in  1837,  have  since  been  confirmed  by  M.  Otho  Struve,  of 
the  celebrated  Pulkova  Observatory. 

Combining  the  determinations  of  these  two  astronomers,  we  find 
the  most  probable  situation  of  the  point  towards  which  the  sun's 


CLUSTERS  OF  STARS. NEBULA. 

motion  is  directed  to  be  as  follows  :  R.  A.  259°  9',  Dec.  N.  34° 
36'.  The  point  in  question  is  situated  in  the  constellation  Hercules, 
near  the  star  u,  (No.  68  in  the  Catalogue  of  the  Astronomical  So- 
ciety,) and  about  10°  from  the  point  first  supposed  by  Herschel. 

594.  O.  Struve  finds  that  for  a  star  situated  at  right  angles  to  the  direction 
of  the  sun's  motion,  and  placed  at  the  mean  distance  of  the  stars  of  the  first 
magnitude,  the  annual  angular  displacement  due  to  the  sun's  motion  is  0".339, 
(with  a  probable  error  of  0".025.)     So  that,  if  we  assume,  according  to  the  best 
determinations,  0".211  for  the  hypothetical  value  of  the  parallax  of  a  star  of  the 
first  magnitude,  it  follows  that  at  the  distance  of  the  star  supposed  the  annual 
motion  of  the  sun  subtends  an  angle  about  once  and  a  half  (1.606)  greater  than 
the  radius  of  the  earth 's^orbit:  which  makes  it  about  150,000,000  of  miles.     This 
is  at  the  rate  of  about  4j  miles  per  second. 

595.  The  above  angle  of  0".339  is  about  the  greatest  annual  displacement  which 
a  star  can  experience  in  consequence  of  the  sun's  motion.    Whence  it  appears  that 
the  whole  of  the  proper  motion  of  any  star  which  is  over  and  above  this  amount 
must  certainly  be  due  to  a  real  motion  in  space.     Thus,  in  the  case  of  the  star  61 
Cygni,  at  least  5"  of  its  annual  proper  motion  (5" .23)  results  from  an  actual  mo- 
tion in  space.     This  is  14.3  times  greater  than  the  parallax  of  this  star,  (0".35.) 
Accordingly  if  we  suppose  the  direction  of  its  motion  to  be  perpendicular  to  its  line 
of  direction  from  the  sun  or  earth,  its  annual  motion  is  14.3  times  greater  than  the 
radius  of  the  earth's  orbit,  or  at  the  rate  of  43  miles  per  second.    As  we  have 
no  means  of  ascertaining  the  actual  direction  of  its  motion,  it  is  impossible  to 
discover  how  much  it  exceeds  this  determination. 

596.  By  comparing  the  particular  motions  presented  by  stars  of  different  class- 
es with  the  motion  of  the  solar  system,  viewed  perpendicularly  at  the  distance  of 
a  star  of  the  first  magnitude,  as  above  given,  it  is  found  that  the  former,  at  the 
mean,  are  2.4  times  greater  than  that  of  the  sun ;  whence  it  follows  that  this  lu- 
minary may  be  ranked  among  those  stars  which  have  a  comparatively  slow  mo- 
tion in  space 

CLUSTERS  OF  STARS.— NEBULAE. 

597.  A  great  number  of  spaces  are  discovered  in  the  heavens 
which  are  faintly  luminous,  and  shine  with  a  pale  white   light. 
These  are  called  Nebula.    Some  are  visible  to  the  naked  eye,  but 
the  greater  number  cannot  be  seen  without  the  aid  of  a  good  tele- 
scope.    On  applying  to  them  telescopes  of  great  power,  they  are 
found  for  the  most  part  to  consist  of  a  multitude  of  small  stars,  dis- 
tinctly separate,  but  very  near  each  other,  and  more  or  less  con- 
densed towards  the  centre. 

598.  There  are  also  clusters  of  stars  in  close  proximity,  dis- 
persed here  and  there  over  the  sphere  of  the  heavens,  which  are 
seen  to  be  such  with  the  naked  eye,  or  with  telescopes  of  only 
moderate  power.     One  of  the  most  conspicuous  of  these  clusters 
is  that  called  the  Pleiades. 

To  the  unaided  sight  it  appears  to  consist  of  six  or  seven  stars, 
but  a  telescope  even  of  moderate  power  exhibits  within  the  space 
they  occupy  fifty  or  sixty  conspicuous  stars.  The  constellation 
called  Coma  Berenices^  is  another  group,  more  diffused,  and  com- 
posed of  larger  stars. 

In  the  constellation  Cancer  there  is  a  luminous  spot,  or  nebula, 
called  Prcesepe,  or  the  bee-hive,  which  a  telescope  of  moderate  power 
resolves  entirely  into  stars.  In  Perseus  is  another  spot  crowded 
with  stars^  which  become  separately  visible  with  a  good  telescope. ' 


224 


OF  THE  FIXED  STARS. 


599.  A  large  number  of  nebulae  are  met  with,  indifferent  parts 
of  the  heavens,  which  offer  no  appearance  of  stars,  even  when  ex- 
amined with  telescopes  of  the  highest  power.  A  very  great  diver- 
sity of  form  and  aspect  obtains  among  them.  One  of  the  most 
prominent  is  that  near  the  star  v  in  Andromeda.  It  is  visible  to  the 
naked  eye,  and  has  often  been  mistaken  for  a  comet.  (See  Fig.  106.N 

Fig.  106. 


600.  The  number  of  nebulae  at  present  known  is  about  3000. 
Although  they  occur  in  almost  every  part  of  the  heavens,  they  are 
the  most  abundant  in  a  zone  perpendicular  to  the  milky  way,  and 
of  about  the  same  breadth,  and  whose  general  direction  is  not  very 
remote  from  that  of  the  equinoctial  colure  ;  and  are  particularly 
numerous  where  it  crosses  the  constellations  Virgo,  Coma  Bereni- 
ces, and  the  Great  Bear.  They  are,  for  the  most  part,  beyond  the 
reach  of  any  but  the  most  powerful  instruments. 

They  are  divided  by  Sir  William  Herschel  into  six  different 
classes,  as  follows  : 

(1.)  Resolved  Nebula ;  that  is,  nebulae  seen  in  the  telescope   to  be  clusters  of 
stars.     Of  these  some  are  globular  in  their  form,  and  others  of  an  irregular  figure 
Fig.  107.  (See  Fig.  107.)     The  latter  are 

less  rich  in  stars,  and  less  con- 
densed towards  the  centre  than 
the  globular  clusters.  They  are 
also  less  definite  in  their  outline. 
It  is  possible  that  they  may  be  in 
the  act  of  condensing,  as  Herschel 
supposed,  and  destined  in  the  pro- 
cess of  ages  to  form  truly  globu- 
lar clusters.  This  idea  seems  to 
be  supported  by  the  fact  of  tho 
occurrence  of  a  regular  gradation 
of  clusters,  from  one  which  seems 
to  be  only  a  space,  of  an  irregular 
and  ill-defined  outline,  somewhat 
more  rich  in  stars  than  the  sur- 
rounding regions,  to  the  perfectly 
defined  and  isolated  globular  clus- 
ter highly  condensed  at  the  cen» 


NEBULA. 


225 


trc.  Globular  clusters  appear  in  telescopes  of  only  moderate  power,  as  small,  round, 
or  oval  nebulous  specks,  resembling  a  comet  without  a  tail.  The  number  of  stars 
which  they  contain  is  to  be  told  only  by  thousands  and  tens  of  thousands ;  although 
their  apparent  size  does  not  exceed  the  y^th  part  of  the  moon's  disc. 

(2.)  Resolvable  Nebulae;  or  such  as  give  indications  that  they  are  clusters  of 
sjars,  and  that  they  are  in  their  nature  resolvable  into  stars,  although  the  power 
of  the  telescope  is  not  yet  sufficient  to  accomplish  this.  In  telescopes  of  the  high- 
est power  they  present  the  same  appearance  as  the  resolved  globular  clusters  in 
telescopes  which  do  not  show  their  individual  stars.  Many  of  them  have  the  as- 

Fig.  108. 


pect  of  these  globular  clusters  just  before  they  are  resolved,  and  which  has  been 
characterized  by  the  phrase  star-dust.  They  are  of  a  round  or  oval  form  ;  and  are 
doubtless  real  clusters  too  distant  to  show  either  their  irregular  edges  or  their  indi- 
vidual stars.  (See  Fig.  108.) 

(3.)  Nebulae  Proper;  or  which  offer  no  appearance  of  stars,  and  are  supposed 
to  be  actual  masses  of  nebulous  matter.     Their  nebulous  constitution  is  inferred, 

Fig.  109. 


Nebula  in  Orion. 

1st,  from  their  unique  appearance,  which  is  often  quite  different  from  that  of  the 
resolvable  nebulae,  and  unlike  what  might  be  supposed  to  arise  from  an  accumula- 
tion of  stars.  (See  Fig.  109.)  2d.  From  their  manifest  physical  connection  with 

29 


OF  THE  FIXED  STARS 


individual  stars  much  superior  to  them  in  brightness.  The  evidence  of  this  physi- 
eal  connection  is  found,  in  some  instances,  in  an  apparent  condensation  upon  the 
in  others  in  the  fact  that  the  substance  of  the  nebula,  whatever  it  may  be, 


star; 


Fig.  110. 


has  apparently  vacated  the  surrounding  regions  of  space  and  accumulated  about 
certain  stars  ;  (see  Fig.  110  ;)  and  in  others  still,  in  the  circumstance  of  its  being 
apparently  drawn  towards  certain  stars.  (See  Fig,  110.)  3.  Another  argument  to 
the  same  point  is,  that  though  many  of  them  are  seen  in  telescopes  of  moderate 
power,  and  some  with  the  naked  eye,  they  are  not  only  not  resolved  into  stars  by 
the  largest  telescopes,  as  other  nebula;  of  the  same  brightness  are,  but  do  not  like 
these  assume  a  different  appearance,  farther  than  that  they  grow  brighter,  as  the 
illuminating  power  of  the  telescope  increases. 

Fig.  111. 


ihey  present  the  greatest  variety  of  forms,  and  occur  in  every  stage  of  apparent 
condensation,  from  rude  amorphous  masses  of  almost  equally  diffused  nebulous 
matter,  to  masses  in  which  the  condensation  has  progressed  so  far  that  a  star  is, 
to  all  appearance,  beginning  to  be  formed  at  the  centre.  The  latter  class  have 
received  a  distinctive  name,  and  will  soon  be  particularly  noticed.  The  condenaa- 


DISTANCE    OF    NEBULJS.  227 


Jion  is  often  going  on  in  the  same  mass  upon  several  lines  or  points ;  and 
are  formed,  presenting,  in  a  regular  gradation,  all  the  varieties  of  appearance, 
which  a  mass,  breaking  up  into  parts  by  condensation  upon  points  or  lines,  would 
assume  down  to  the  time  of  complete  separation.  The  nebulas  in  which  the  con- 
densation appears  to  be  upon  one  point  or  line,  are  round  or  oval  in  their  figure. 
But  some  are  long  and  spindle-shaped,  while  others  are  perfectly  circular.  A  very 
few  of  the  round  nebulae  are  annular.  (Fig.  111.)  A  conspicuous  example  of 
this  singular  class  of  nebulae  may  be  seen  with  a  telescope  of  moderate  power 
midway  between  the  stars  /?  and  y  Lyrae.  By  far  the  greater  portion  of  the  nebulas 
proper  are  round. 

(4.)  Planetary  Nebula ;  or  nebulae  which  have  an  appearance  similar  to  the  plan- 
ets ;  being  round,  of  an  equable  light  throughout,  and  often  perfectly  definite  in  their 
outline.  (See  Fig.  112.)     The  uniformity  of  their  light  seems          fig.  112, 
to  indicate  that  it  proceeds  altogether  from  the  surface  of  some 
spherical  body ;  and  therefore  that  the  body,  if  it  be  a  collec- 
tion of  nebulous  matter,  is  of  the  form  of  a  spherical  shell :  or 
else  that  it  is  derived  from  a  bed  of  stars  of  uniform  thick- 
ness.   The  latter  supposition  seems  to  be  the  more  probable  one, 
and  is  moreover  now  known  to  be  true  in  some  instances ;  for 
Lord  Rosse  has  succeeded  in  resolving  one  of  the  planetary  ne- 
bulae of  Sir  J.  Herschel's  catalogue,  (viz.  Fig.  49  ;)  and  has  discovered  another 
(Fig.  45  of  the  catalogue)  to  be  an  annular  nebula. 

The  largest  planetary  nebula  occurs  in  the  Swan,  and  is  nearly  15' in  diameter. 

(5.)  Stellar  Nebula;  that  is,  nebulae  so  much  condensed  at  the  centre  as  to  offer 
the  appearance  of  a  star  there  seen  through  the  surrounding  nebulous  mass.    (See 

Fig.  112.« 


Fig.  112.°)  In  some  instances  the  condensation  is  gradual,  in  others  sudden.  A 
good  example  of  stellar  nebulae  occurs  to  the  south  of  the  star  .3  Ursa?  Majoris. 

(6.)  Nebulous  Stars;  or  stars  distinctly  seen  to  be  such,  surrounded  by  their 
nebulous  atmospheres.  On  the  supposition  of  progressive  condensation,  they 
would  seem  to  be  stellar  nebulae  in  a  more  advanced  state.  (See  Fig.  Il2.ffl) 

Among  stellar  nebulae  and  nebulous  stars  there  exist  particular  nebulae  in  every 
stage  of  apparent  condensation,  from  the  slightest  appearance  of  a  star  at  the  cen- 
tre to  a  perfect  star  surrounded  with  the  faintest  nebulous  haze.  (See  Note  XIX.) 

DISTANCE  AND  MAGNITUDE  OF  NEBULA. 

601.  Herschel  undertook  to  estimate  the  distance  of  resolved 
nebulae,  by  noting  the  space-penetrating  power  of  the  telescope 
which  first  succeeded  in  revealing  their  distinct  stars.     According 
to  his  determinations,  therefore,  the  most  remote  of  the  resolved 
nebulae  are  at  the  same  distance  as  the  most  remote  of  the  isolated 
stars  discerned  in  his  large  telescope  ;  and  thus  at  least  1000  times 
the  distance  of  the  nearest  and  brightest  stars.     The  others  are 
distributed  at  the  same  variety  of  distance  as  the  telescopic  stars. 

602.  As  to  the  actual  dimensions  of  these  clusters,  if  we  sup- 
pose the  distance  of  none  of  them  to  be  more  than  1000  times  the 
distance  of  a  star  of  the  first  magnitude,  a  globular  cluster  whose 


22S  OF   THE    FIXED    STARS. 

apparent  diameter  is  10'  cannot  have  a  real  diameter  of  more  than 
three  stellar  intervals.  At  a  distance  some  5  times  greater,  such 
a  cluster  would  contain  several  thousand  stars  as  remote  from  each 
other  as  is  the  nearest  fixed  star  from  our  sun. 

603.  It  is  to  be  supposed  that  the  resolvable  nebulae  are  in  gene- 
ral posited  beyond  the  region  of  resolved  nebulae  and  visible  stars. 
The  nearest  of  them  are  on  the  very  confines  of  this  region.     We 
may  form  some  estimate  of  the  probable  distance  of  the  most  re- 
mote of  these  objects,  by  calculating  how  much  farther  a  cluster, 
ascertained  as  above,  (601,)  to  be  at  1000  times  the  distance  of 
the  nearest  star,  and  which  is  just  discerned  as  a  whitish  speck  by 
a  telescope  of  the  space-penetrating  power  20,  would  have  to  be 
removed  to  have  the  same  appearance  in  a  telescope  whose  power 
is  200,  (which  is  less  than  the  power  of  the  largest  telescope.)     It 
is  plain  that  it  would  have  to  be  removed  1 0  times  farther,  or  to 
1 0,000  times  the  distance  of  the  nearest  isolated  stars. 

This  calculation  supposes,  however,  that  the  number  of  stars  in  the  most  remote 
resolvable  nebulee  is  no  greater  than  in  the  most  distant  resolved  nebulae.  If  we 
suppose  the  number  to  be  greater  in  any  ratio,  the  distance  will  be  increased  in  the 
proportion  of  the  square  root  of  the  same  ratio.  Thus  suppose  the  number  of  stars 
in  the  remotest  resolved  nebulae  to  be  10,000,  and  that  the  most  distant  of  the  re- 
solvable nebulas  contains  a  number  1000  times  greater,  or  10,000,000,  (which  is 
not  far  from  the  estimate  of  the  number  of  stars  in  the  stratum  of  the  milky  way,) 
(566).  The  distance  calculated  above  will  be  increased  about  30  times,  that  is, 
will  be  no  less  than  300,000  times  a  stellar  interval — a  distance  so  enormous  that 
light  would  employ  1,000,000  of  years  in  traversing  it.  Some  astronomers  make 
the  probable  distance  of  stars  of  the  lowest  magnitude  about  three  times  less  than 
we  have  taken  it,  which  would  make  the  distance  just  calculated  less  in  the  same 
proportion.  Herschel,  on  the  other  hand,  makes  it  about  twice  as  great.  If  the 
bed  of  stars  to  which  our  sun  belongs  were  viewed  at  this  distance,  it  would  sub- 
tend an  angle  of  about  10',  and  appear  about  —^th  of  the  size  of  the  moon's  disc. 
It  seems  probable,  a  priori,  that  other  similar  beds  of  stars,  to  that  in  which  our 
sun  is  posited,  occur  in  the  profundities  of  space.  If  this  is  the  case  they  must 
then  be  visible  at  the  enormous  distance  just  stated,  unless  there  be  a  limit  to -the 
known  law  of  the  propagation  of  light. 

604.  It  appears  then  that  clusters  of  stars  are  distributed  through- 
out space  at  every  variety  of  distance,  from  that  of  stars  of  about 
the  4th  magnitude  to  an  unknown  limit  beyond  the  reach  of  the 
most  powerful  telescopes  :  and  that  the  telescope  succeeds  in  dis- 
tinctly resolving  only  those  which  are  posited  within  the  region  of 
the  isolated  stars  discernible  through  it.     The  more,  distant  ones 
appear  in  it  as  spots  of  nebulous  light,  and  occupy  the  fields  of 
space  which  extend  from  say  1000  times  the  distance  of  stars  of 
the  first  magnitude  to  at  least  10,000  times  this  distance. 

605.  As  respects  the  nebulae  proper,  we  may  form  some  esti- 
mate of  the  distance  of  some  of  them  by  noting  the  magnitude  of 
the  stars  with  which  they  seem  to  be  connected.     In  this  way,  for 
example,  it  is  found  that  the  remarkable  nebula  in  Orion  occupies 
the  interval  between  stars  of  the  3d  and  8th  magnitudes.     It  is 
probable  that  some  of  these  objects,  which  give  no  indications  of  a 
physical  connection  with  stars,  lie  beyond  the  region  of  known 


STRUCTURE  OF  THE  MATERIAL  UNIVERSE.  229 

stars,  but  we  have  no  means  of  obtaining  even  the  remotest  ap- 
proximation to  the  distance  of  individuals  among  them. 

606.  A  mere  speck  in  the  heavens,  at  the  distance  of  the  stars, 
as  viewed  through  a  good  telescope,  is  as  large  as  the  earth's  or- 
bit :  accordingly  the  collections  of  nebulous  matter  which  occur  in 
the  heavens,  in  the  regions  of  the  stars,  must  have,  at  least,  as 
great  a  superficial  extent  as  the  orbit  of  the  earth.     Many  of  them 
must  be  vastly  larger.     For  example,  the  nebula  in  Andromeda 
(599)  is  two-thirds  of  the  apparent  size  of  the  moon's  disc.     Its 
actual  extent  cannot  be  less  than  365,000  times  that  of  the  earth's 
orbit,  or  1000  times  that  of  the  whole  solar  system.  (See  Note  XX.) 

607.  The  matter  of  the  smallest  of  these  nebulae  may  be  ex- 
ceedingly subtile,  and  yet  be  sufficient  in  quantity  to  condense  into 
a  body  as  large  and  as  dense  as  the  sun ;  for  it  appears,  by  calcu- 
lation, that  if  the  matter  of  the  sun  were  to  expand  so  as  to  fill  the 
space  enclosed  within  the  earth's  orbit,  it  would  be  about  45,000 
times  rarer  than  the  air. 

STRUCTURE  OF  THE  MATERIAL  UNIVERSE— NEBULAR 
HYPOTHESIS. 

608.  In  view  of  the  facts  which  have  now  been  presented,  it 
will  be  seen  that  the  great  prominent  feature  in  the  structure  of  the 
universe  is  the  arrangement  of  the  stars  in  detached  beds.     Thus 
our  starry  firmament  is  one  immense  bed  of  stars,  in  which  occur 
a  great  number  of  subordinate  clusters  or  beds,  so  that,  in  fact,  it 
appears  to  be  chiefly  made  up  of  more  or  less  detached  and  con- 
densed groups  of  stars.     Exterior  to  this  stratum,  as  far  as  the 
telescope  penetrates  into  the  abyss  of  space,  are  seen  other  beds, 
apparently  similar,  for  the  most  part,  to  those  which  occur  within 
the  stratum  itself.     But  some  that  are  seen,  it  is  not  improbable, 
are  other  firmaments  constructed  on  the  same  vast  scale  as  that  of 
the  milky  way,  and  at  a  distance  from  it  of  200  or  300  times  its 
own  diameter,  (603.)     Leaving  these  out  of  view,  the  others,  al- 
though occurring  here  and  there  in  almost  every  direction,  beyond 
the  stratum  of  the  milky  way,  seem  to  be,  the  great  majority  of  them, 
disposed  in  a  stratum  of  unknown  extent,  crossing  the  stratum  of  the 
milky  way  nearly  at  right  angles  ;  or  rather,  if  the  milky  way  was 
correctly  gauged  by  Herschel,  surrounding  it,  without  anywhere 
touching  it.     These  beds  are  of  a  great  variety  of  forms.     But  the 
greater  number  of  them  are  generally  supposed  to  be  spherical,  or 
nearly  so.     Some  which  have  been  supposed  to  have  this  form, 
may,  perhaps,  be  circular  or  elliptical  strata,  or  of  the  form  of 
spherical  segments,  more  condensed  towards  the  centre,  and  seen 
either  perpendicularly  in  their  true  form,  or  obliquely,  so  as  to  have 
their  longer  axis  foreshortened.     Planetary  nebulae  may  be  such 
strata,  which  are  not  condensed  towards  the  centre  :  and  annular 
nebulae,  the  same,  in  which  there  is  a  deficiency  of  stars  at  the 
central  parts.     (See  Note  XXI.) 


230  Of   THE    FIXED    STARS 

609.  The  discoveries  that  have  been  made  in  the  heavens  seem 
then  to  point  to  this  great  truth,  viz.,  that  the  plan  upon  which 
the  universe  has  been  fashioned,  is  that  of  an  ascending  scale  of 
systems,  from  isolated  suns  with  their  attendant  systems  of  planets, 
to  the  stupendous  whole  which  fills  the  eye  of  the  Infinite  Creator. 

But  the  indications  are  that  the  work  of  creation  is  still  in  progress.  Dispersed 
through  the  realms  of  space,  as  we  have  seen  (p.  226),  are  immense  masses  of  some 
sort  of  nebulous  matter,  which  in  their  various  stages  of  condensation  upon  one 
or  more  points  or  lines,  seem  to  present  in  sibylline  leaves  the  whole  history  of  the 
progressive  creation  of  existing  worlds  and  systems  of  worlds,  and  at  the  same  time 
to  picture  forth  the  accomplishment  of  a  similar  destiny  on  the  part  of  these  masses 
themselves. 

610.  The  theory  that  worlds  have  been  and  are   still  being 
slowly  evolved  from  primordial  nebulous  masses  by  the  gradual 
operation  of  the  general  forces  and  properties  which  the  Creator 
has   either  permanently  imparted   to   matter,    or   is   incessantly 
renewing  in  it,  is  called  the  Nebular  Hypothesis.     Its  author  is 
Sir  William  Herschel.     But  Laplace,  by  undertaking  to  trace  in 
detail  the  progress  of  the  creation  of  the  solar  system,  has  still 
more  effectually  stamped  his  name  upon  it  than  the  author  him- 
self.    The  great  arguments  which  are  urged  in  its  support  are  the 
following  : 

(1.)  That  there  is  a  multitude  of  shining  nebulas  masses  now 
scattered  throughout  space,  each  of  sufficient  extent  to  furnish  the 
materials  of  a  world,  and  some  perhaps  of  systems  of  worlds. 

(2.)  That  these  masses  present  a  long  unbroken  gradation,  from 
a  mass  "  without  form  and  void"  to  a  perfect  star :  that  is,  all  the 
various  states  in  which  a  single  nebulous  mass  would  be  during 
the  vast  period  that  it  occupies  in  condensing  from  its  first  rude 
formless  state  into  a  finished  globe. 

(3.)  That  the  universe,  as  it  is,  in  both  the  general  and  particu- 
lar features  of  its  structure,  may  be  shown  to  be  a  natural  me- 
chanical consequence  of  the  hypothesis  in  question. 


PART   III. 

OF  THE  THEORY  OF  UNIVERSAL  GRAVITATION. 


CHAPTER  XIX. 

i 

OF    THE    PRINCIPLE    OF   UNIVERSAL    GRAVITATION. 

611.  IT  is  demonstrated  in  treatises  on  Mechanics,  that  if  a  body 
move  in  a  curve  in  such  a  manner  that  the  areas  traced  by  the 
radius -vector  about  a  fixed  point,  increase  proportionally  to  the 
times,  it  is  solicited  by  an  incessant  force  constantly  directed  to- 
wards this  point. 

The  following  is  a  geometrical  proof  of  this  principle.  Conceive  the  orbit  to  be 
a  polygon  of  an  infinite  number  of  sides.  Let  ABCD  (Fig.  113)  be  a  portion  of 


Fig.  113. 


it ;  and  S  the  fixed  point  about  which  the  radius- 
vector  describes  areas  proportional  to  the  times, 
or  equal  areas  in  equal  times.  Since  the  impulses 
are  only  communicated  at  the  angular  points 
A,  B,  C,  D,  &c.,  of  the  polygon,  the  motion  will 
be  uniform  along  each  of  the  sides  AB,  BC,  CD, 
&c. :  and  since  we  may  suppose  the  times  of  de- 
scribing  these  sides  to  be  equal,  we  shall  have  the 
triangular  area  SAB  equal  to  the  triangular  area 
SBC,  and  SBC  equal  to  SCD,  &c.  Produce  AB 
and  make  Be  equal  to  AB,  which  may  be  taken  to 
represent  the  velocity  along  AB  ;  and  join  Cc.  Cc 
will  be  parallel  to  the  line  of  direction  of  the  impulse 
that  lakes  effect  at  B.  Upon  SB  let  fall  the  per- 
pendiculars Am,  en,  Cr.  Then,  since  AB  =  Be, 
Am  =  en  ;  and  since  the  equivalent  triangles 
SAB,  SBC,  have  a  common  base  SB,  Am  =  Cr. 
It  follows,  therefore,  that  en  =  Cr,  and  conse- 
quently, that  Cc  is  parallel  to  BS.  The  impulse 
which  the  body  receives  at  B  is  therefore  directed 
from  B  towards  S.  In  the  same  manner  it  may 
be  shown  that  the  impulse  which  it  receives  at  C  is  directed  from  C  towards  S. 
The  line  of  direction  of  the  force  passes,  therefore,  in  every  position  of  the  body, 
through  the  point  S. 

Now,  by  Kepler's  first  law,  the  areas  described  by  the  radii- 
vectores  of  the  planets  about  the  sun,  are  proportional  to  the  times. 
It  follows  therefore  from  this  law,  that  each  planet  is  acted  upon 
by  a  force  which  urges  it  continually  towards  the  sun,. 

This  fact  is  technically  expressed  by  saying  that  the  planets 
gravitate  towards  the  sun,  and  the  force  which  urges  each  planet 
towards  the  sun  is  called  its  Gravity,  or  Force  of  Gravity,  towards 
the  sun. 

612.  It  is  also  proved  by  the  principles  of  Mechanics,  that  if  a 
body,  continually  urged  by  a  force  directed  to  some  point,  describe. 


232 


OF  THE  PRINCIPLE  OF  UNIVERSAL  GRAVITATION. 


an  ellipse  of  which  that  point  is  a  focus,  the  force  by  which  it  is 
urged  must  vary  inversely  as  the  square  of  the  distance. 

Thus,  let  ABG  (Fig.  114)  be  the 
supposed  elliptic  orbit  of  the  body, 
C  A  and  CB  its  semi-axes,  and  S  the 
focus  towards  which  the  force  is 
constantly  directed.  Also  let  P  be 
one  position  of  the  body,  PR  a  tan- 
gent to  the  orbit  at  P ;  and  draw 
RQ  parallel  to  PS,  Qwu,  HI,  and 
CD,  parallel  to  PR,  Qx  perpendicu- 
»  lar  to  SP,  PF  perpendicular  to  CD, 
and  join  S  and  Q.  CP  and  CD  are 
semi-conjugate  diameters.  Denote 
them,  respectively,  by  A'  and  B' ; 
and  denote  the  semi-axes,  CA  and 
CB,  by  A  and  B.  Since  HI  is  par- 
allel to  PR,  and,  by  a  well-known 


G 


property  of  the  ellipse,  the  angle  RPS  is  equal  to  the  angle  HPT,  PH  is  equal  to 
PI :  and  since  HC  =  SC,  and  CE  is  parallel  to  HI,  E  is  the  middle  of  SI.  We 
have,  therefore, 


2  2 

Now  the  force  at  P  is  measured  by  2Pu  ;  and  we  may  state  the  proportion 

A' 
PK  :  PC  : :  PE :  PC : :  A  :  A' ;  which  gives  Pt?  =  P«  -^. 

By  the  equation  of  the  ellipse  referred  to  its  centre  and  conjugate  diameters, 
PG  and  DL, 


If  we  regard  Q  as  indefinitely  near  to  P,  then  QM  =  Qu,  and  Go  =  2CP  =  2A' ; 
and  therefore 

But  Qtt:Qx::PE:PF::CA:PF: 

and,  by  analytical  geometry, 

CD  X  PF  =  C  A  X  CB,  or,  CA  :  PF  : :  CD :  CB  : :  B' :  B. 

Hence         Qu :  Qx : :  B' :  B,  Qu2:  Qi2 : :  B'2 :  B2,  and  Q«2  =  Q?  -2? 


Substituting  in  equation  (a),     Q*2-™~  =  T~- 2Ptt  5  whence  Qa:2  =  --— .  2Pw. 

U  A.  A 

Now  triangular  area  SQP  =  k  =  SP  X  -~-  ;  whence  Qx  =  —  .   Substituting, 

SP2 

there  results 


SP2 


—  .4*2     1..  CD- 
B2  gp2 


To  compare  the  intensities  of  the  force  at  different  points  of  the  orbit,  we  must 
take  the  values  of  2PM,  by  which  they  are  measured,  for  the  same  interval  of  time. 
On  this  supposition  k  is  constant,  and  therefore  the  force  is  inversely  proportional 
to  the  square  of  the  distance  SP. 

It  therefore  follow?  from  Kepler's  second  law,  viz. :  that  the 
planets  describe  ellipses  having  the  centre  of  the  sun  at  one  of 


233 

their  foci  ;  that  the  force  of  gravity  of  each  planet  towards  the 
sun  varies  inversely  as  the  square  of  the  distance  from  the  sun's 
centre  . 

613.  By  taking  into  view  Kepler's  third  law,  it  is  proved  that  it 
is  one  and  the  same  force,  modified  only  by  distance  from  the  sun, 
which  causes  all  the  planets  to  gravitate  towards  him,  and  retains 
them  in  their  orbits.  This  force  is  conceived  to  be  an  attraction 
of  the  matter  of  the  sun  for  the  matter  of  the  planets,  and  is  called 
the  Solar  Attraction. 

To  deduce  this  consequence  from  Kepler's  third  law,  let  t,  t',  denote  the  periodic 
times  of  any  two  planets  ;  r,  r',  their  distances  from  the  sun  at  any  assumed  point 
of  time  ;  k,  k',  the  areas  described  by  them  in  any  supposed  unit  of  time  ;  and  A,  B, 
and  A',  B',  the  semi-axes  of  their  elliptic  orbits.  Then  kt,  k't,  will  be  equal  to  the 
areas  of  the  entire  orbits  ;  which  are  also  measured  by  n-AB,  jrA'B'. 
Thus  kt  :  k't'  :  :  AB  :  A'B',  and  kW  :  k'2f2  :  :  A2fi2  :  A'2B'2. 

But,  by  Kepler's  third  law,          t?  :  t'2  :  :  A3  :  A'3. 

B2    B'2 

Dividing,  and  reducing,  ft2  :  A;'2  :  :  —  -  :  —  —  : 

.  xx      A. 

that  is,  the  squares  of  the  areas  described  in  equal  times  are  as  the  parameters  of 
the  orbits. 

Now,  let  /,  /,  denote  the  forces  soliciting  the  two  planets.     Then,  by  equation 
(I),  Art.  612, 


From  which  it  appears  that  the  planets  are  solicited  by  a  force  of  gravitation 
towards  the  sun,  which  varies  from  one  planet  to  another  according  to  the  law  of 
the  inverse  square  of  their  distance. 

614.  The  motions  of  the  satellites  are  in  conformity  with  Kep 
ler's  laws  ;  hence,  the  planets  which  have  satellites  are  endued 
with  an  attractive  force  of  the  same  nature  with  that  of  the  sun. 

615.  The  existence  of  a  similar  attractive  power  in  each  of  the 
planets  that  are  devoid  of  satellites,  is  proved  by  the  fact  that  the 
observed  inequalities  of  their  motions,  and  of  those  of  the  other 
planets,  may  be  shown  upon  this  supposition  to  be  necessary  con- 
sequences of  the  attractions  of  the  planets  for  each  other. 

616.  In  like  manner  the  inequalities  in  the  motions  of  the  satel- 
lites and  their  primaries,  show  that  the  satellites  possess  the  same 
property  of  attraction  as  the  sun. 

617.  We  learn  from  the  motions  produced  by  the  action  of  the 
sun  and  planets  upon  each  other,  that  the  intensities  of  their  at- 
tractive forces  are,  at  the  same  distance,  proportional  to  their 
masses,  and  that  the  whole  attraction  of  the  same  body  for  differ- 
ent bodies,  is,  at  the  same  distance,  proportional  to  the  masses  of 
these  bodies.     From  which  we  may  infer  that  a  mutual  attraction 
exists  between  the  particles  of  bodies,  and  that  the  whole  force  of 
attraction  of  one  body  for  another,  is  the  result  of  the  attractions 

30 


234         THEORY  OF  THE  ELLIPTIC  MOTION  OF  THE  PLANETS. 

of  its  individual  particles.  Moreover,  analysis  shows,  that  in  or- 
der that  the  law  of  attraction  of  the  whole  body  may  be  that  of  the 
inverse  ratio  of  the  square  of  the  distance,  this  must  also  be  the 
law  of  attraction  of  the  particles.  The  fact,  as  well  as  the  law  of 
the  mutual  attraction  of  particles,  is  also  revealed  by  the  tides  and 
other  phenomena  referable  to  such  attraction.  •* 

618.  The  celestial  phenomena  compared  with  the  general  laws 
of  motion,  conduct  us  therefore  to  this  great  principle  of  nature ; 
namely,  that  all  particles  of  matter  mutually  attract  each  other  in 
the  direct  ratio  of  their  masses,  and  in  the  inverse  ratio  of  the 
squares  of  their  distances.   This  is  called  the  principle  of  Univer- 
sal Gravitation.     The  theory  of  its  existence  was  first  promul- 
gated by  Sir  Isaac  Newton,  and  is  hence  often  called  Newton's 
Theory  of  Universal  Gravitation.     The  force  which  urges  the 
particles  of  matter  towards  each  other  is  called  the  Force  of  Grav- 
itation^ or  the  Attraction  of  Gravitation. 

619.  In  the  following  chapters  our  object  will  be  to  develope  the 
most  important  effects  of  the  principle  of  gravitation  thus  arrived  at 
by  induction.     The  perfect  accordance  that  will  be  observed  to 
obtain  between  the  deductions  from  the  theory  of  universal  gravi- 
tation and  the  results  of  observation,  will  afford  additional  confir- 
mation of  the  truth  of  the  theory. 


CHAPTER  XX. 

THEORY  OF  THE  ELLIPTIC  MOTION  OF  THE  PLANETS. 

620.  LET  the  attraction  of  the  unit  of  mass  of  the  sun  for  the 
unit  of  mass  of  a  planet,  at  the  unit  of  distance,  be  designated  by  1 . 
The  whole  attraction  exerted  by  the  sun  upon  the  unit  of  mass,  at 
the  same  distance,  will  then  be  expressed  by  the  mass  of  the  sun 
(M) ;  or,  in  other  words,  by  the  number  of  units  which  its  mass 
contains.  And  the  attraction  F,  at  any  distance  r,  will  result  from 

the  proportion  M  :  F  : :  r2 : 12,  which  gives  F=  -j.  This,  in  the  lan- 
guage of  Dynamics,  is  the  Accelerating  Force  soliciting  the  planet. 
As  —2  expresses  the  attraction  of  the  sun  for  a  unit  of  mass  of 
the  planet,  its  attraction  for  the  entire  mass  m  of  the  planet  will  be 
expressed  by  m  -^.  This  is  the  moving  force  of  the  planet,  and 
since  it  is,  at  the  same  distance,  proportional  to  the  mass  of  the 


REVOLUTION  ABOUT  THE  CENTRE  OF  GRAVITY. 


231 


planet,  the  velocity  due  to  its  action  is  the  same,  whatever  may  be 
the  mass. 

621.  The  planet  has  also  an  attraction  for  the  sun,  as  well  as  the 
sun  for  the  planet,  and  the  expression  for  its  attractive  force,  or  for 

the  accelerating  force  animating  the  sun,  will  obviously  be  -3.  The 

sun  will  then  tend  towards  the  planet,  as  the  planet  towards  the 
sun.  But,  if  the  two  bodies  were  to  set  out  from  a  state  of  rest, 
the  velocity  of  the  planet  would  be  as  many  times  greater  than  the 
velocity  of  the  sun,  as  the  mass  of  the  sun  is  greater  than  that  of 
the  planet.  For  the  velocity  of  the  planet  would  be  to  that  of  the 
sun  as  the  attractive  force  of  the  sun  is  to  the  attractive  force  of 

M    m 

the  planet,  that  is,  as  -^  :  -^,  or  as  M  :  m. 

As  the  attraction  of  the  particles  of  the  sun  and  planet  are  mu- 
tual and  equal,  the  attraction  of  the  planet  for  the  entire  mass  of 
the  sun  must  be  equal  to  the  attraction  of  the  sun  for  the  entire 
mass  of  the  planet. 

622.  The  sun  and  any  planet  revolve  about  their  common  cen- 
tre of  gravity. 

To  show  this,  we  would  remark,  in  the  first  place,  that  it  is  a 
principle  of  Mechanics  that  the  mutual  actions  of  the  different 
members  of  a  system  of  bodies  cannot  affect  the  state  of  the  centre 
of  gravity  of  the  system.  This  is  called  the  Principle  of  the 
Preservation  of  the  Centre  of  Gravity.  It  follows  from  it  that 
the  common  centre  of  gravity  of  the  sun  and  any  planet  is  at  rest, 
unless  it  has  a  motion  of  translation  in  common  with  the  two  bo- 
dies, imparted  by  a  force  extraneous  to  the  system.  As  we  are 
concerned  at  present  only  with  the  relative  motion  of  the  sun  and 
planet,  such  motion  of  translation,  if  it  does  exist,  may  be  left 
out  of  account.  Now,  let  S  (Fig.  115)  be  the  Fig.  115. 

sun,  and  P  any  planet,  supposed  for  the  mo- 
ment to  be  at  rest.  If  neither  of  the  two  bo- 
dies should  receive  a  velocity  in  a  direction 
oblique  to  PS,  the  line  of  their  centres,  they 
would  move  towards  each  other  by  virtue  of 
their  mutual  attraction,  and  meet  at  C  their 
common  centre  of  gravity.*  But,  if  the  body 
P  have  a  projectile  velocity  given  to  it  in  any 
direction  Pt,  inclined  to  the  line  PS,  it  is  sus- 
ceptible of  proof  that  its  motion  relative  to  the 
sun  may  be  in  an  ellipse,  as  is  observed  to 
be  the  case  with  the  planets. 

Now,  while  the  planet  moves  in  space,  the  line  of  the  centres 


*  The  common  centre  of  gravity  of  two  bodies  lies  on  the  line  joining  their  cen- 
tres, and  divides  this  line  into  parts  inversely  proportional  to  the  masses  of  the 
bodies. 


236         THEORY  OF  THE  ELLIPTIC  MOTION  OF  THE  PLANETS. 

of  the  planet  and  sun  must  continually  pass  through  the  stationary 
position  of  the  centre  of  gravity  ;  and  therefore,  when  the  planet  has 
advanced  to  any  point  p,  the  sun  will  have  shifted  its  position  to 
some  point  s  on  tne  line  pC  prolonged.  Moreover,  as  the  two  bo- 
dies mutually  gravitate  towards  each  other,  the  paths  of  each  in 
space  will  be  continually  concave  towards  the  other  body,  and 
therefore  also  towards  the  centre  of  gravity  C,  which  is  constantly 
in  the  same  direction  as  the  other  body.  Since  the  planet  per- 
forms a  revolution  around  the  sun,  the  sun  and  planet  must  each 
continue  to  move  about  the  point  C  until  they  have  accomplished 
a  revolution  and  returned  to  the  line  PCS.  Also,  as  the  distance 
PS  of  the  two  bodies  will  be  the  same  at  the  end  as  at  the  begin- 
ning of  the  revolution,  as  well  as  the  ratio  of  their  distances  PC 
and  SC  from  the  centre  of  gravity,  they  will  return  to  the  posi- 
tions P,  S,  from  which  they  set  out,  and  will  therefore  move  in 
continuous  curves. 

Moreover,  these  curves  are  similar  to  the  apparent  orbit  described  by  P  around 
S.  For,  draw  Sp1  parallel  and  equal  to  sp,  and  join  Pp  and  Ss.  Then,  since 
sC  :  Cp  :  :  SC  :  CP,  Pp  is  parallel  to  Ss  ;  and  therefore  Pp  produced  passes 
through  p1.  Whence,  CP :  Cp : :  SP  :  Sp'.  Moreover,  the  angle  PCp  =  PS/.  It 
follows,  therefore,  that  the  area  PCp  is  similar  to  the  area  PS//  ;  and  thus  that  the 
orbit  of  P  around  C  is  similar  to  the  apparent  orbit  of  P  around  S.  The  latter  is 
known  from  observation  to  be  an  ellipse.  The  former  is  therefore  also  an  ellipse. 

As  the  distances  of  the  sun  and  planet  from  their  common  cen- 
tre of  gravity  are  constantly  reciprocally  proportional  to  their 
masses,  the  orbit  of  the  sun  will  be  exceedingly  small  in  compari- 
son with  the  orbit  of  the  planet. 

623.  If  to  both  the  sun  and  planet  there  should  be  applied  a 

force  equal  to  the  accelerating  force  of  the  sun,  -^,  (621),  but  in  an 

opposite  direction,  the  sun  would  be  solicited  by  two  forces  that 
would  destroy  each  other,  but  the  planet  would  now  be  urged 
towards  the  sun  remaining  stationary,  with  the  accelerating  force 

— TJ — ,  or  a  force  the  intensity  of  which  was  equal  to  the  sum  of 

the  intensities  of  the  attractive  forces  of  the  sun  and  planet,  at  the 
distance  of  the  planet.  Now,  the  application  of  a  common  force 
will  not  alter  the  relative  motion  of  the  two  bodies.  Hence,  in  in- 
vestigating this  motion,  we  are  at  liberty  to  conceive  the  sun  to  be 
stationary,  if  we  suppose  the  planet  to  be  solicited  by  the  accelera- 
ting force  — -g — .  As  the  mass  of  the  sun  is  very  much  greater 

than  that  of  any  planet,  but  little  error  will  be  committed  in  neg- 
lecting the  attraction  of  the  planet,  and  taking  into  account  only 

the  sun's  action  — §. 
r 

624.  Analysis  makes  known  the  general  laws  of  the  motion  of 
a  body,  when  impelled  by  a  projectile  force,  and  afterwards  contin- 


GENERAL  PROBLEM  OF  MOTION  OF  REVOLUTION.  237 

ually  attracted  towards  the  sun's  centre  by  a  force  varying  inverse- 
ly as  the  square  of  the  distance.  We  learn  by  it  that  the  body 
will  necessarily  describe  some  one  of  the  conic  sections  around 
the  sun  situated  at  one  of  its  foci.  We  learn,  also,  that  the  na- 
ture of  the  orbit,  as  well  as  the  length  of  the  major  axis,  is  wholly 
dependent,  for  any  given  distance  of  the  planet,  upon  the  intensity 
of  the  projectile  force,  but  that  the  position  of  the  axis,  and  the  ec- 
centricity of  the  orbit,  depend  also  upon,  the  angle  of  projection 
(that  is,  the  angle  included,  at  the  commencement  of  the  motion, 
between  the  line  of  direction  of  the  projectile  force  and  the  radius- 
vector.)  As  to  the  relative  intensity  of  the  projectile  force  neces 
sary  to  the  production  of  each  one  of  the  conic  sections,  a  certain 
intensity  of  force  will  produce  a  parabola ;  any  less  intensity,  an 
ellipse  or  circle  ;  and  any  greater,  an  hyperbola. 

625.  If  the  velocity  that  would  at  a  given  distance  be  imparted 
by  the  sun's  attraction  in  a  second  of  time,  which  is  the  measure 
of  its  intensity  at  the  given  distance,  be  found,  and  also  the  dis- 
tance of  a  planet  at  any  time,  as  well  as  its  velocity  and  the  angle 
made  by  the  direction  of  its  motion  with  the  radius-vector,  the  form, 
dimensions,  and  position  of  the  planet's  orbit  can  be  computed. 
This  is  to  determine  the  orbit  a  priori.     The  practice  has  been, 
however,  to  determine  the  various  elements  of  a  planet's  orbit  by 
observation,  (as  already  described,  Chap.  VII.) 

The  elements  being  known,  the  equations  of  the  elliptic  mo- 
tion, investigated  on  the  principles  of  Mechanics,  serve  to  make 
known  the  position  and  velocity  of  the  planet  at  any  time.  (The 
investigation  of  these  equations  may  be  found  in  the  Encyclopaedia 
Metropolitana,  Article  Physical  Astronomy,  page  653,  in  the  Me- 
canique  Elementaire  de  Francoeur,  and  in  many  other  similar 
works.)* 

626.  The  physical  theory  of  the  motion  of  a  satellite  around  its 
primary  is  obviously  the  same  as  that  of  the  motion  of  a  planet 
around  the  sun. 

627.  According  to  the  principle  of  the  preservation  of  the  centre 
of  gravity  (622),  the  centre  of  gravity  of  the  whole  solar  system 
must  either  be  at  rest,  or  have  a  motion  of  translation  in  space  in 
common  with  the  system,  resulting  from  the   action  of  a  foreign 
force.     We  have  already  seen  (593)  that  it  has  been  ascertained 
from  observation,  that  it  is  in  fact  in  motion. 

628.  The  sun  and  planets  revolve  around  their  common  centre 
of  gravity.     The  path  of  the  sun's  centre  results  from  the  joint  ac- 
tion of  all  the  planets,  and  is  a  complicated  curve.    As  the  quan 
tity  of  matter  in  all  the  planets  taken  together  is  very  small,  com 
pared  with  that  in  the  sun,  (less  than  Tio-,)  the  extent  of  the  curve 
described  by  the  centre  of  the  sun  cannot  be  very  great.     It  is 

*  The  equations  are  the  same  with^hose  deduced  directly  from  Kepler's  laws  ol 
the  planetary  motions. 


£38         THEORY  OP  THE  ELLIPTIC  MOTION  OF  THE  PLANETS. 

found  by  computation,  that  the  distance  between  the  sun's  centre 
and  the  centre  of  gravity  of  the  system  can  never  be  equal  to  the 
sun's  diameter. 

629.  It  is  demonstrated  in  treatises  on  Mechanics,  that  if  foreign 
forces  act  upon  a  system  of  bodies,  the  centre  of  gravity  of  the  sys- 
tem will  move  just  as  the  whole  mass  of  the  system  concentrated 
at  the  centre  of  gravity  would  move,  under  the  action  of  the  same 
forces.     It  follows  from  this  principle,  that  from  the  attraction  of 
the  sun  for  a  primary  planet  and  its  satellites,  their  common  cen- 
tre of  gravity  will  revolve  around  the  sun,  just  as  the  whole  quan- 
tity of  matter  in  the  planet  and  its  satellites  concentrated  at  this 
point  would,  under  the  influence  of  the  same  attraction.  Moreover 
the  same  considerations  which  show  that  the  sun  and  planets  re- 
volve about  their  common  centre  of  gravity,  will  also  show  that  a 
primary  planet  and  its  satellites  revolve  about  their  common  centre 
of  gravity.     It  appears,  therefore,  that  in  the  case  of  a  planet 
which  has  satellites,  it  is  not,  strictly  speaking,  the  centre  of  the 
planet  that  moves  agreeably  to  the  first  and  second  laws  of  Kepler, 
but  the  common  centre  of  gravity  of  the  planet  and  its  satellites  ; 
the  planet  and  satellites  revolving  around  the  centre  of  gravity,  as 
it  describes  its  orbit  about  the  sun. 

630.  It  may  be  worth  while  here  to  remark,  that  the  revolution 
of  the  earth  around  the  common  centre  of  gravity  of  the  earth  and 
moon,  occasions  an  inequality,  both  of  longitude  and  latitude,  in 
the  apparent  motion  of  the  sun.    It  is,  however,  exceedingly  small, 
for  the  reason  that  the  distance  of  the  earth's  centre  from  the  cen- 
tre of  gravity  is  very  short,  in  comparison  with  the  distance  of  the 
sun.     The  mass  of  the  earth  is  to  that  of  the  moon  as  80  to  1, 
while  the  distance  of  the  moon  is  to  the  radius  of  the  earth  as  60 
to  1  :  it  follows,  therefore,  that  the  common  centre  of  gravity  of  the 
earth  and  moon  lies  within  the  body  of  the  earth. 

631.  It  appears  also  from  the  physical  investigation  of  the  ellip- 
tic motion  of  the  planets,  that  Kepler's  third  law  is  not  rigorously 
true.     In  consequence  of  the  action  of  the  planets  upon  the  sun, 
the  ratio  of  the  periodic  times  of  the  different  planets  depends  upon 
the  masses  of  the  planets,  as  well  as  their  distances  from  the  sun. 
If  p  and  p'  be  the  periodic  times  of  any  two  of  the  planets,  a  and  a' 
their  mean  distances  from  the  sun's  centre,  and  m  and  m'  their 
quantities  of  matter,  that  of  the  sun  being  denoted  by  1,  then,  dis- 
regarding the  actions  of  the  other  planets, 


. 
1+w     1+m' 

As  m  and  m'  are  very  small  fractions,  the  error  resulting  from  their 
omission  will  be  very  small.     If  we  omit  them,  we  shall  have 

/:p"::a3:a*. 
which  is  Kepler's  third  law. 


INVESTIGATION  OF  THE  DISTURBING  FORCES. 


239 


CHAPTER  XXI. 

THEORY  OF  THE  PERTURBATIONS  OF  THE  ELLIPTIC  MOTION  OF  THE 
PLANETS  AND  OF  THE  MOON. 


116. 


632.  WE  have,  in  a  previous  chapter,  given  a  general  idea  of  the  mode  of 
determining,  from  theory  and  observation  combined,  the  law  and  amount  of 
the  perturbations  or  inequalities  of  the  lunar  and  planetary  motions.  We  pro- 
pose now  to  give  some  insight  into  the  nature  and  manner  of  operation  of  the 
disturbing  forces,  and  will  commence  with  the  perturbations  of  the  moon  pro- 
duced by  the  action  of  the  sun. 

633.  We  have  already  (283)  shown  how  the  intensity  and  direction  of 
the  disturbing  force  of  the  sun,  in  any  given  position  of  the  moon  in  its  orbit, 
may  be  determined.    Let  us  now  derive  the  disturbing  forces  that  take  eifect 
in  the  three  directions  in  which  the  motion  of  the  moon  can  be  changed ; 
namely,   in   the   direction    of   the   radius- 
vector,    of  the   tangent   to   the    orbit,  and 

of  the  perpendicular  to  its  plane.  Let  E 
(Fig.  116)  be  the  earth,  M  the  moon,  and 
S  the  sun.  Let  the  force  exerted  by  the 
sun  upon  the  moon  be  decomposed  into  two 
forces,  one  acting  along  the  line  MS'  par- 
allel to  ES,  and  the  other  from  M  towards 
E.  If  the  component  along  MS'  were  equal 
to  the  force  exerted  by  the  sun  upon  the 
earth,  the  motion  of  the  moon  about  the 
earth  would  not  be  changed  by  the  action 
of  these  two  forces.  Hence,  the  difference 
between  them  will  be  the  disturbing  force  in 
the  direction  MS'.  The  component  along 
ME  is  another  disturbing  force.  It  is  called 
the  Addititious  Force,  because  it  tends  to 
increase  the  gravity  of  the  moon  towards  the 
earth.  The  disturbing  force  along  MS'  will 
generally  be  inclined  to  the  plane  of  the 
orbit,  and  may  be  decomposed  into  three 
forces,  one  in  the  direction  of  the  tangent, 
another  in  the  direction  of  the  radius-vec- 
tor, and  a  third  in  the  direction  of  the  per- 
pendicular to  the  plane.  The  first  men- 
tioned component  is  called  the  Tangential 

Force  ;  the  second  is  called  the  Ablatitious  Force ;  and  the  third  we  shall  call 
the  Perpendicular  Force. 

The  actual  disturbing  force  in  the  direction  of  the  radius-vector  is  equal  to 
the  difference  between  the  addititious  and  ablatitious  forces,  and  is  called  the 
Radial  Force.  This  and  the  tangential  and  perpendicular  forces  constitute 
the  disturbing  forces,  the  direct  operation  of  which  is  to  be  considered. 

634.  To  obtain  general  analytical  expressions  for  these  forces,  let  the  dis- 
tance of  the  sun  from  the  earth  (which  for  the  present  we  shall  suppose  to 
be  constant)  be  denoted  by  a,  and  the  distances  of  the  moon  from  the  earth 
and  sun,  respectively,  by  y  and  z.     Also  let  F  =  the  force  exerted  by  the 
earth  upon  the  moon,  P  =  the  force  exerted  by  the  sun  upon  the  earth,  and 
Q  =  the  force  exerted  by  the  sun  upon  the  moon.     Then,  if  we  denote  the 


SB" 


240         PERTURBATIONS  OP  ELLIPTIC  MOTION  OP  THE  MOON. 


mass  of  the  earth  by  1,  and  take  m  to  stand  for  the  mass  of  the  sun,  we  shall 
have,  (620,) 

F--L  P--  Q-- 

JC       — —    •""       '  '  4    A       —  0J   ^t   Q    * 

y"  a*  z* 

Let  the  force  Q  be  represented  by  the  line  MS  (Fig.  116)  ;  and  let  its 
component  parallel  to  ES,  or  MS'  =  R,  and  its  component  along  the  radius- 
vector,  or  ME  =  T. 

Q  :  T  : :  MS  :  ME  ;  or,  -^  :  T  :  :  z  :  y. 

Whence,          addititious  force  T  =  ^  .  .  .  (130). 
In  a  similar  manner  we  obtain 


The  disturbing  force  in  the  direction  of  the  sun 


Now,  let  a,  &  y,  denote  the  angles  made  by  the  line  MS',  respectively,  with 
the  tangent,  the  radius-vector,  and  the  perpendicular  to  the  plane  of  the  orbit, 
and  we  shall  have  for  the  components  of  the  disturbing  force  R  —  P,  along 
these  lines  ; 


cos  a 


/I         1  \ 

tangential  force  =  ma  I  ~^s  —  "^"1 


ablatitious  force  =  ma 


perpendicular  force  =ma  I  TF j   I  cos  y 


(132) 


(133); 


(134). 


Fig.  117. 


Combining  equation  (133)  with  equation  (130) 
we  obtain  for  the  radial  force, 

radial  force = my— §-  — ma  I  — ^  I  cos  /?. 

635.  The  obliquity  of  the  orbit  of  the  moon 
to  the  plane  of  the  ecliptic,  affects  but 'very 
slightly  the  value  of  the  tangential  and  radial 
forces.  If  we  leave  it  out  of  account,  or  sup- 
pose the  moon's  orbit  to  lie  in  the  plane  of  the 
ecliptic,  we  shall  have  (Fig.  117)  j3==S/ML 
=  SEM  the  elongation  of  the  moon  =  0,  and 
a  =  complement  of  0,  which  gives 


tang,  force  =  ma  I  -—  —  — - 


(135); 


rad.  force=my mat ,  ) cos  q>  (136). 

y  z3  \z*  aj 

636.  Equation  (134)  may  be  transformed 
.into  another,  which  is  better  adapted  to  the 
purposes  we  have  in  view.  Let  MK  (Fig. 
116)  represent  the  perpendicular  to  the  plane 
of  the  moon's  orbit,  MF  the  intersection  of  the  plane  SMK  with  the  plane 
of  the  moon's  orbit,  and  SI,  IF  the  intersections  of  a  plane  passing  through 


INVESTIGATION  OF  THE  DISTURBING  FORCES.  241 

S  and  perpendicular  to  EN,  the  line  of  nodes,  with  the  plane  of  the  ecliptic 
and  the  plane  of  the  orbit.  SF  will  be  perpendicular  to  both  IF  and  MF 
Denote  SIF,  the  inclination  of  the  orbit  to  the  ecliptic,  by  I,  SEN  the  angu- 
lar distance  of  the  sun  from  the  node  by  N,  and  SE  ajid  SM  by  a  and  z>  as 
before. 

Now,  in  equation  (134)  Y  stands  for  the  angle  S'MK,  but  S'MK  =  SMK, 
(nearly,)  and 

SF 
cos  SMK  =  sin  SMF  =  ^-. 

SF  =  SI  sin  SIF,  and  SI  =  SE  sin  SEI  ; 
whence  SF  =  SE  sin  SEI  sin  SIF  =  a  sin  N  sin  I  : 

substituting. 

a  sin  N  sin  I      a  sin  N  sin  I 

cos  y  =  cos  SMK  =  -  —  —  -  =  --  . 
oM  z 

Thus  we  have 

/I         1  \  a  sin  N  sin  I 
perpen.  force  =  mal  •—  --  -  I  —    --   —  ...  (137). 

637.  The  variable  z  may  be  eliminated  from  equations  (135),  (136),  and 
(137),  and  other  equations  obtained,  involving  only  the  variables  y  and  0.  Let 
ML  (Fig.  116)  be  drawn  through  the  place  of  the  moon  perpendicular  to  ES. 
Then,  using  the  same  notation  as  in  the  preceding  articles, 

LS  =  z  (nearly),  EL  =  EM  cos  LEM  =  y  cos  0. 
But  LS  =  SE  —  EL; 

whence  z  =  a  —  y  cos  0,  and  z9  =  a3  —  3a2y  cos  0  : 

neglecting  the  terms  containing  the  higher  powers  of  y  than  the  first,  as  they 
are  very  minute,  y  being  only  about  3-^  a. 

I  __  1  __  1        3y  cos  0 
"^"^a3  —  3aaycos0  =="^"~*    ~~tf       ' 
neglecting  all  the  terms  of  the  quotient  that  involve  higher  powers  of  y  than 

the  first.     Substituting  this  value  of  —  in  equation  (135),  we  obtain, 


my  cos  A  sn  d> 
;  ^  :  tangential  force  =  -  -  --  ^-  -  £•  ; 

or,  (App.  For.  13), 

3my  sin  20 
tangential  force  =  —  -  —  -  —    .  .  .  (138). 

Making  the  same  substitution  in  equation  (136),  and  neglecting  the  term  con- 
taining ya,  there  results, 

,.  .  ,  my  (I  —  3cos'0) 

radial  force  =  —       —  -3  —      —  ; 

or,  (App.  For.  9), 

,.  ,  f  .my  (1+3  cos  20) 

radial  force  =  --  -  -  x—  3  -  —   .  .  .  (139). 

In  equation  (137)  we  have  to  substitute,  besides,  the  value  of  z,  viz.  a  —  y 
cos  0  ;  then  dividing  and  neglecting  as  before,  we  have 

3wiy  cos  0 
perpen.  force  =  -  -3  -  sin  N  sin  I  ...  (140.) 

638.  If  the  disturbing  forces  retained  constantly  the  same  intensity  and  di- 
rection, the  result  would  be  a  continual  progressive  departure  from  the  ellip- 
tic place  ;  but,  in  point  of  fact,  these  forces  are  subject  to  periodical  changes 
of  intensity  and  direction  from  several  causes,  from  which  results  a  compeu- 

31 


242         PERTURBATIONS  OF  ELLIPTIC  MOTION  OF  THE  MOON. 


eation  of  effects,  and  an  eventual  return  to  the  elliptic  place.  The  causes  of 
the  variation  of  the  disturbing  forces  are  : 

(1.)  The  revolution  of  the  moon  around  the  earth. 

(2.)  The  elliptic  form  of  the  apparent  orbit  of  the  sun. 

(3.)  The  elliptic  form  of  the  orbit  of  the  moon. 

(4.)  The  inclination  of  the  two  orbits. 

As  the  variations  of  the  radial  and  tangential  forces,  resulting  from  the  in- 
clination of  the  orbits,  are  very  minute,  we  shall  leave  them  out  of  account, 
and  in  the  consideration  of  the  effects  of  these  forces  shall,  for  the  sake  of 
simplicity,  regard  the  orbits  as  lying  in  the  same  plane. 

The  first  mentioned  circumstance  is  the  most  prominent  cause  of  variation, 
and  gives  rise  to  the  more  conspicuous  perturbations.  The  other  two  serve 
to  modify  the  variations  of  the  forces  resulting  from  the  first,  and  occasion 
each  a  distinct  set  of  periodical  perturbations. 

639.  Let  us  now  investigate,  in  succession,  the  effects  of  eacn  of  the  dis- 
turbing forces,  commencing  with  the  tangential  force.  The  tangential  force 
takes  effect  directly  upon  the  velocity  of  the  moon  in  its  orbit ;  and  as  its  line 
of  direction  does  not  pass  through  the  earth,  it  disturbs  the  equable  descrip- 
tion of  areas.  It  also  affects  the  radius-vector  indirectly,  by  changing  the 
centrifugal  force.  To  understand  the  detail  of  its  action  we  must  inquire  in- 
to the  variations  which  it  undergoes. 

If  we  regard  y  as  constant  in  the  expression  for  the  tangential  force,  (eqna. 
138,)  which  amounts  to  considering  the  moon's  orbit  as  circular,  the  expres- 
sion will  become  equal  to  zero  when  sin  2<p  =  0,  and  will  have  its  maximum 
value  when  sin  20=  1.  It  will  also  change  its  sign  with  sin  20.  It  appears, 
therefore,  that  the  tangential  force  is  zero  in  the  syzigies  and  quadratures, 
where  it  also  changes  its  direction,  and  that  it  attains  its  maximum  value  in 


Fig.  118. 


the  octants.  It  will  be  seen,  on  inspect 
ing  Fig.  118,  that  it  will  be  a  retarding 
force  in  the  first  quadrant,  (AB)  Accord- 
ingly, it  will  be  an  accelerating  force  in 
the  second,  a  retarding  force  again  in  the 
third,  and  an  accelerating  force  again  in 
the  fourth. 

This  will  also  appear  upon  considering 
the  direction  of  the  disturbing  force  par- 
allel to  the  line  of  the  centres  of  the  sun 
and  earth,  in  the  various  quadrants.  In 
the  nearer  half  of  the  orbit  the  sun  tends 
to  draw  the  moon  away  from  the  earth, 
and  the  force  in  question  is  directed  to- 
wards the  sun.  In  the  more  remote  half 
the  sun  tends  to  draw  the  earth  away  from  the  moon,  but  we  may  regard  it, 
instead,  as  urging  the  moon  from  the  earth  by  the  same  force  ;  for  the  rela- 
tive motion  will  be  the  same  on  this  supposition.  In  the  part  of  the  orbit 
supposed,  then,  the  disturbing  force  under  consideration  will  be  directed  from 
the  sun,  as  represented  in  Fig.  118. 

640.  It  appears,  then,  that  the  tangential  force  will  alternately  retard  and 
accelerate  the  motion  of  the  moon  during  its  passage  through  the  different 
quadrants,  and  that  the  maximum  of  velocity  will  occur  in  the  syzigies.  A,  C, 
where  the  accelerating  force  becomes  zero,  and  the  minimum  of  velocity  in 
the  quadratures,  B,  D,  where  the  retarding  force  becomes  zero.  On  the  sup- 
position that  the  orbit  is  a  circle,  the  arcs  AB,  BC,  CD,  and  DA,  would  be 
equal,  and  the  retardation  of  the  velocity  in  one  quadrant  would  be  compen- 
sated for  by  an  equal  acceleration  in  the  next,  and  at  the  close  of  a  synodic 
revolution  the  velocity  of  the  moon  would  be  the  same  as  at  its  commence- 
ment. As  the  velocity  is  greatest  in  the  syzigies  and  least  in  the  quadratures, 
and  as  the  degree  of  retardation  is  the  same  as  that  of  acceleration,  the  menu 


EFFECTS  OF  THE  TANGENTIAL  FORCE.  243 

motion*  must  have  place  in  the  octants.  Now,  as  the  moon  moves  from  the 
syzigy  A  with  a  motion  greater  than  the  mean  motion,  her  true  place  will  be 
in  advance  of  her  mean  place,  and  will  become  more  and  more  so  till  she 
reaches  the  octant,  where  the  true  motion  is  equal  to  the  mean.  The  dif- 
ference between  the  true  and  mean  place  will  then  be  the  greatest ;  for  after 
that,  the  true  motion  becoming  less  than  the  mean,  the  mean  place  will  ap- 
proach nearer  to  the  true,  till  at  the  quadrature  they  coincide.  Beyond  B, 
the  true  motion  still  continuing  less  than  the  mean,  the  mean  place  will  be  in 
advance  of  the  true,  and  the  separation  will  increase  till  at  the  octant  the 
true  motion  has  attained  to  an  equality  with  the  mean  motion,  after  which,  the 
mean  motion  being  the  slowest,  the  true  place  will  approach  the  mean  till  at 
the  syzigy  C  they  again  coincide.  Corresponding  effects  will  take  place  in 
the  two  remaining  quadrants.  We  perceive,  therefore,  that  the  tangential 
force  produces  an  inequality  of  longitude,  which  attains  to  its  maximum  posi- 
tive and  negative  value  in  the  octants,  and  is  zero  in  the  syzigies.  This  is  the 
inequality  known  in  Plane  Astronomy  by  the  name  of  Variation,  (296.) 

641.  Let  us  now  inquire  into  the  modifications  of  the  effects  of  the  tangen- 
tial force,  that  result  from  the  elliptic  form  of  the  sun's  orbit.     Suppose  that 
at  the  moment  when  the  moon  sets  out  from  conjunction  the  sun  is  in  the 
apogee  of  its  orbit :  then  it  is  plain  that,  during  the  whole  revolution  of  the 
moon,  the  sun's  disturbing  force  would  be  on  the  increase  by  reason  of  the 
diminution  of  the  sun's  distance,  and  that,  in  consequence,  the  retardation  in 
the  first  quadrant  would  be  less  than  the  acceleration  in  the  second,  and  the 
retardation  in  the  third  less  than  the  acceleration  in  the  fourth.     So  that, 
when  the  moon  had  again  come  round  into  conjunction,  the  acceleration  would 
have  over-compensated  the  retardation.     This  kind  of  action  would  go  on  so 
long  as  the  sun  approached  the  earth  ;  but  when  it  had  passed  the  perigee  of 
its  orbit,  and  began  to  recede  from  the  earth,  the  reverse  effect  would  take 
place,  and  a  retardation  of  the  moon's  orbitual  motion  would  happen  each 
revolution.    If  the  anomalistic  revolution  of  the  sun  was  an  exact  multiple  of 
the  synodic  revolution  of  the  moon,  the  acceleration  in  each  revolution  of  the 
moon  during  the  passage  of  the  sun  from  the  apogee  to  the  perigee  of  its  or- 
bit, would  be  compensated  for  by  an  equivalent  retardation  in  the  revolution 
of  the  moon  answering  to  the  same  distance  of  the  sun  in  its  passage  from  the 
apogee  to  the  perigee ;  and  the  velocity  of  the  moon  would  be  the  same  at 
the  close  of  an  anomalistic  revolution  of  the  sun  as  at  its  commencement.   But 
as  this  relation  does  not,  in  fact,  subsist  between  the  anomalistic  revolution 
of  the  sun  and  the  synodic  revolution  of  the  moon,  a  compensation  between 
the  accelerations  and  retardations,  answering  to  the  different  revolutions  of 
the  moon,  will  not  be  effected  until  conjunctions  shall  have  occurred  at  every 
variety  of  distance  of  the  sun  in  each  half  of  its  orbit.     Since  the  anomalistic 
and  synodic  revolutions  are  incommensurable,  the  sun  will  be,  in  reality,  in 
every  variety  of  position  in  its  orbit  at  the  time  of  conjunction,  in  process  of 
time  ;  so  that  eventually  the  original  velocity  in  conjunction  will  be  regained, 
It  appears,  therefore,  that  the  variation  of  the  moon's  motion  from  one  revo- 
lution to  another,  occasioned  by  the  elliptic  form  of  the  sun's  orbit,  is  periodic. 
Its  period  will  be  the  interval  of  time  in  which  the  moon  will  perform  a  cer- 
tain number  of  synodic  revolutions,  while  the  sun  performs  a  certain  number 
of  anomalistic  revolutions.    Avoiding  unnecessary  precision,  we  find  it  to  con- 
sist of  but  a  moderate  number  of  years. 

642.  We  have  next  to  consider  the  consequences  of  the  elliptic  form  of 
the  moon's  orbit.     We  remark,  in  the  first  place,  that,  the  orbit  being  an 
ellipse,  the  areas  AEB,  BEG,  CED,  and  DEA,  (Fig.  118,)  will  be  unequal, 
and  therefore,  by  the  laws  of  elliptic  motion,  the  arcs  AB,  BC,  CD,  and  DA, 
will  be  described  in  unequal  times.     It  follows  from  this,  that  the  retardation 

»  The  expressions,  mean  motion,  true  motion,  mean  place,  true  place,  are  here 
to  be  understood  only  in  relation  to  the  perturbation  under  consideration. 


244          PERTURBATIONS  OF  ELLIPTIC  MOTION  OF  THE  MOON 

in  the  first  quadrant  will  not  be  exactly  compensated  by  the  acceleration  in 
the  second,  and  that  the  retardation  in  the  third  will  not  be  exactly  compen- 
sated by  the  acceleration  in  the  fourth.  Therefore,  at  the  end  of  the  synodic 
revolution  the  moon  will  have  an  excess  or  deficiency  of  velocity.  Its  mean 
motion  will  then  vary  from  one  revolution  to  another,  by  reason  of  the  ellip- 
ticity  of  its  orbit.  This  variation  will  be  periodic,  like  that  just  considered", 
and  for  similar  reasons.  The  excess  or  deficiency  of  velocity  at  the  close  of 
any  one  revolution,  will  in  time  be  compensated  by  an  equal  deficiency  or 
excess  occurring  at  the  close  of  another  revolution,  when  the  sun  has  a  cer- 
tain different  position  with  respect  to  the  perigee  of  the  moon's  orbit. 

643.  We  pass  now  to  the  consideration  of  the  action  of  the  radial  force. 
The  direct  general  effect  of  the  radial  force,  is  an  alteration  in  the  intensity 
of  the  moon's  gravity  towards  the  earth,  and  in  its  law  of  variation.     Its 
specific  effects  are  periodical  variations  in  the  magnitude,  eccentricity,  and 
position  of  the  orbit.     As  it  is  directed  towards  the  earth,  it  will  not  disturb 
the  equable  description  of  areas.     To  discover  the  variations  of  this  force 
we  have  only  to  discuss  the  general  analytical  expression  for  it,  already  in- 
vestigated.    It  is, 

my  (I — 3  cos3^) 
radial  force  =  -^ -3 — . 

We  shall  have  radial  force  =  0,  when  1  —  3  cos2  9  =  0,  or  when  cos 
0  =  ±  V?-  This  value  of  cos  ^  answers  to  four  points  lying  on  either  side 
of  the  quadratures,  and  about  35°  distant  from  them.  When  cos  ^  is  numeri- 
cally greater  than  ^/±  the  result  will  be  negative,  and  when  it  is  less  than 
•/^  the  result  will  be  positive.  It  follows,  therefore,  that  the  radial  force 
increases  the  gravity  of  the  moon  in  the  quadratures,  and  for  about  35°  on 
each  side  of  them,  and  that  during  the  remainder  of  a  synodic  revolution  it 
diminishes  it. 

When  the  moon  is  in  quadratures,  cos  0=0,  and 

radial  force  =—....  (141). 
In  the  syzigies,  we  have  cos  #  =  i  1,  which  gives 
radial  force  = *jr  .  .  .  (142). 

It  appears,  then,  that  the  diminution  of  the  moon's  gravity  in  the  syzigies 
is  double  of  its  increase  in  the  quadratures. 

We  learn  also  from  equations  (141)  and  (142),  that  the  radial  force  in  the 
quadratures  and  syzigies  varies  directly  as  the  distance  ;  from  which  we  con- 
clude that  the  gravity  of  the  moon  varies  at  these  points  by  a  different  law 
from  that  of  the  inverse  squares.  In  the  quadratures  the  gravity  will  be  in- 
creased most  at  the  greatest  distance,  where  it  is  the  least ;  and  thus  it  will 
vary  in  a  less  rapid  ratio  than  the  square  of  the  distance.  In  the  syzigies  it 
will  be  diminished  most  at  the  greatest  distance,  or  where  it  is  the  least ;  and 
accordingly,  at  these  points  it  will  vary  in  a  more  rapid  ratio  than  the  square 
of  the  distance. 

644.  An  easy  investigation,  with  the  aid  of  the  differential  calculus,  proves 

that  the  mean  diminution  of  the  moon's  gravity  from  the  sun's  action  is  -^-j ; 

r  representing  in  this  case  the  mean  distance  of  the  moon  from  the  earth. 
The  value  of  this  expression  is  readily  found  to  be  equal  to  about  the  360th 
part  of  the  whole  gravity  of  the  moon  to  the  earth. 

In  consequence  of  this  diminution,  the  moon  must  describe  her  orbit  at  a 
greater  distance  from  the  earth,  with  a  less  angular  velocity,  and  in  a  longei 
time,  than  if  she  were  acted  on  only  by  the  attraction  of  the  earth. 

645.  The  radial  force  of  the  sun  alters  the  eccentricity  of  the  moon's  orbit 


EFFECTS  OF  THE  RADIAL  FORCE.  245 

and  differently  in  different  revolutions  of  the  moon,  according  to  the  position 
of  the  line  of  syzigies  with  respect  to  the  line  of  apsides.  When  these  lines 
are  coincident  the  eccentricity  is  increased. 
For,  suppose  PMAN  (Fig.  119)  to  be  the 
elliptic  orbit  of  the  moon  that  would  be 
described  under  the  influence  of  a  force 
varying  inversely  as  the  square  of  the  dis- 
tance. In  going  from  the  apogee  to  the 
perigee,  the  gravity  will  increase  in  a 
greater  ratio  than  that  of  the  inverse 
square  of  the  distance  ;  the  true  orbit  will 
therefore  fall  within  the  ellipse,  and  the 
perigean  distance  (EP')  will  be  less  than 
for  the  ellipse.  Consequently,  the  eccen- 
tricity will  increase  so  much  the  more  as 
the  major  axis  diminishes.  On  the  other  hand,  in  going  from  the  perigee  to 
the  apogee,  the  gravity  will  decrease  in  a  greater  ratio  than  the  inverse  square 
of  the  distance,  and  the  moon  will  consequently  recede  farther  from  the  earth 
than  if  the  orbit  described  was  an  ellipse.  Therefore,  in  this  half  of  the  or- 
bit the  eccentricity  will  also  be  increased.  When  the  apsides  are  in  quadra- 
tures the  eccentricity  will  be  diminished  ;  for  the  gravity  will  then  vary  from 
the  apogee  to  the  perigee,  and  from  the  perigee  to  the  apogee,  in  a  less  ratio 
than  that  of  the  inverse  squares ;  and  therefore  the  results  will  be  contrary 
to  those  just  obtained.  The  eccentricity  will  have  its  maximum  value  when 
the  apsides  are  in  syzigies,  and  its  minimum  when  they  are  in  quadratures  ; 
for,  in  every  other  position  of  the  line  of  apsides  with  respect  to  the  line  of 
syzigies,  the  radial  force  in  the  apogee  and  perigee  will  be  less  than  in  these 
positions,  (equa.  139.)  and  therefore  alter  less  the  proportional  gravity  of  the 
moon  in  the  apogee  and  perigee.  It  is  evident,  from  the  gradual  decrease  ot 
the  radial  force  as  we  recede  from  the  syzigies  and  quadratures,  that  the  ec- 
centricity will  continually  diminish  in  the  progress  of  the  apsides  from  the 
syzigies  to  the  quadratures,  and  that  it  will  continually  increase  from  the 
quadratures  to  the  syzigies. 

The  change  in  the  eccentricity  of  the  moon's  orbit,  thus  produced,  will  be 
attended  with  a  corresponding  change  in  the  equation  of  the  centre,  and  thus 
of  the  longitude.  And  this  change  is  the  conspicuous  inequality  of  the  moon, 
known  by  the  name  of  Evection,  (296.) 

646.  The  radial  force  also  produces  a  motion  of  the  line  of  apsides.  If  the 
moon  was  only  acted  upon  by  the  attraction  of  the  earth  its  orbit  would  be  an 
ellipse,  and  the  motion  from  one  apsis  to  another,  or,  in  other  words,  from 
one  point  where  the  orbit  cuts  the  radius-vector  at  right  angles  to  the  other, 
would  be  180°.  In  point  of  fact,  however,  the  gravity  due  to  the  earth's 
attraction  is  constantly  either  diminished  or  increased  by  the  radial  disturbing 
force  of  the  sun,  and  therefore  its  true  orbit  must  continually  deviate  from 
the  ellipse  that  would  be  described  under  the  sole  action  of  the  earth's  attrac- 
tion. When  from  the  action  of  this  force  there  is  a  diminution  of  the  moon's 
gravity,  she  will  continually  recede  from  the  ellipse  in  question,  her  path  will 
be  less  bent,  and  she  must  therefore  move  through  a  greater  angular  distance 
before  the  central  force  will  have  deflected  her  course  into  a  direction  at  right 
angles  to  the  radius-vector.  Accordingly,  she  will  move  through  a  greater 
angular  distance  than  180°  in  going  from  one  apsis  to  another,  and  thus  the 
apsides  will  advance.  On  the  other  hand,  when  the  same  force  increases  the 
moon's  gravity,  her  path  will  fall  within  the  ellipse,  its  curvature  will  be  in- 
creased, and  therefore  it  will  be  brought  to  intersect  the  radius-vector  at  right 
angles  at  a  less  angular  distance.  In  this  case,  therefore,  the  apsides  will 
move  backward.  Now,  we  have  shown  (643)  that  the  radial  disturbing  force 
of  the  sun  alternately  diminishes  and  increases  the  moon's  gravity  to  the  earth. 
It  follows,  therefore,  that  the  motion  of  the  apsides  will  be  alternately  direct 


246         PERTURBATIONS  OF  ELLIPTIC  MOTION  OF  THE  MOON. 


and  retrograde  ;  but  since,  as  has  been  shown,  (643,)  the  diminution  subsists 
during  a  longer  part  of  the  moon's  revolution,  and  is  moreover  greater  than 
the  increase,  the  direct  motion  will  exceed  the  retrograde,  and  therefore  in 
an  entire  revolution  the  apsides  wjll  advance. 

647.  The  observed  motion  of  the  apsides  of  the  moon's  orbit  is  not,  how- 
ever, wholly  produced  by  the  radial  disturbing  force.     It  is  in  part  due  to  the 
action  of  the  tangential  force.     This  force  alters  the  centrifugal  force  of  the 
moon,  and  thus  changes  its  gravity  towards  the  earth,  at  the  same  time  with 
the  radial  force. 

648.  The  elliptic  form  of  the  sun's  orbit  is  the  occasion  of  a  change  in 
the  radial  force,  from  which  results  a  perturbation  of  longitude  called  the  An- 
nual Equation,  (296.)     The  mean  diminution  of  the  moon's  gravity,  arising 

from  the  action  of  the  sun,  or  the  mean  radial  force,  is  equal  to  -—  8,  (644.) 

Hence  this  diminution  is  inversely  proportional  to  the  cube  of  the  sun's  dis- 
tance from  the  earth.  Therefore,  as  the  sun  approaches  the  perigee  of  its 
orbit,  its  distance  from  the  earth  diminishing,  the  mean  diminution  of  the 
moon's  gravity  to  the  earth  will  increase,  and  consequently  the  moon's  dis- 
tance from  the  earth  will  become  greater,  and  its  motion  slower,  than  it  other- 
wise would  be.  The  contrary  will  take  place  while  the  sun  is  moving  from 
the  perigee  to  the  apogee. 

649.  The  disturbing  force  perpendicular  to  the  plane  of  the  moon's  orbit, 
produces  a  tendency  in  the  moon  to  quit  that  plane,  from  which  there  results 
a  change  in  the  position  of  the  line  of  the  nodes,  and  a  change  in  the  inclina- 
tion of  the  plane  of  the  orbit  to  that  of  the  ecliptic.    If  we  examine  the  gene- 
ral expression  for  this  force,  viz  : 


3my  cos 
perpen.  force  =  --  3 


. 
sin  N  sm  I, 


Fig.  120. 


we  see  that  for  any  given  values  of  N  and  I,  it  will  be  zero  in  the  quadra- 
tures, and  have  its  greatest  value  in  the  syzigies ;  and  that  it  will  change  its 
direction  in  the  quadratures,  lying,  in  the  nearer  half  of  the  orbit,  on  the 
same  side  of  its  plane  as  the  sun,  and  in  the  more  remote  half,  on  the  opposite 
side.  We  perceive  also  that  it  will  be  zero  for  every  value  of  <p,  or  for  every 
elongation  of  the  moon,  when  the  angle  N  is  zero,  that  is,  when  the  sun  is  in 
the  plane  of  the  orbit ;  and  will  attain  its  maximum,  for  any  given  elongation, 
when  the  line  of  direction  of  the  sun  is  perpendicular  to  the  line  of  nodes. 
It  will  also  be  the  less,  other  things  being  the  same,  the  smaller  is  the  incli- 
nation I. 

650.  Now  let  NM'R  (Fig.  120)  repre 
sent  the  orbit  of  the  moon,  and  S  the  sun, 
supposed  stationary,  the  line  of  the  nodes 
being  in  quadratures ;  and  let  L,  L'  be  the 
points  of  the  orbit  90°  distant  from  the 
nodes.  The  direction  of  the  force,  in  the 
various  points  of  the  orbit,  is  indicated  by 
the  arrows  drawn  in  the  figure.  When  the 
moon  is  at  any  point  M'  between  L  and  the 
descending  node  N',  she  will  be  drawn  out 
of  the  plane  in  which  she  is  moving  by  the 
disturbing  force.  MTC',  and  compelled  to 
move  in  such  a  line  as  M'J'.  The  node  N' 
will  therefore  retrograde  to  some  point  n'. 
When  she  is  at  any  point  M,  moving  from 
the  ascending  node  N  towards  L,  her  course 
will  be  changed  to  the  line  Mt,  lying,  like 
the  line  M'Z',  below  the  orbit,  which  being  produced  backward,  meets  the 
plane  of  the  ecliptic  in  some  point  n,  behind  N.  The  nodes,  therefore,  retro- 


EFFECTS  OF  THE  PERPENDICULAR-  FORCE. 


247 


121 


grade  in  this  position  of  the  moon,  as  well  as  in  the  former.  When  the  moon 
is  in  Ihe  half  N'L'N  of  the  orbit,  lying  below  the  ecliptic,  the  absolute  direc- 
tion of  the  disturbing  force  will  be  reversed,  and  thus  its  tendency  will  be  the 
same  as  before,  namely,  to  draw  the  moon  towards  the  ecliptic.  It  follows, 
therefore,  that  throughout  this  half  of  the  orbit,  as  in  the  other,  the  motion  ot 
the  nodes  will  be  retrograde.  Accordingly,  when  the  nodes  are  in  quadra- 
tures, or  90°  distant  from  the  sun,  they  will  retrograde  during  every  part  of 
the  revolution  of  the  moon. 

651.  Suppose  the  sun  now  to  be  fixed  on  the  line  of  nodes,  or  the  nodes  to 
be  in  syzigies.     In  this  case  the  perpendicular  force  will  be  zero,  (649,)  and 
therefore  there  will  be  no  disturbance  of  the  plane  of  the  moon's  orbit. 

652.  Next,  let  the  situation  of  the  sun  be  intermediate  between  the  two 
just  considered,  as  represented  in  Figs.  120  and  121.     The  eifect  of  the  dis 
turbing  force  will  be  the  same  as  in  the  first  situation  from  the  quadrature  q 
(Fig.  120)  to  the  node  N',  and  from  the  quadrature  q1  to  the  node  N.     But 
throughout  the  arcs  Ny,  N'y',  the  direction  of  the  force,  and  therefore  the 
effects,  will  be  reversed.    The  node  will  then  retrograde,  as  before,  while  the 
moon  moves  over  the  arcs  ^N'  and  y'N,  and  advance  while  she  is  in  the  arcs 
N^,  N'^'.     But  as  the  force  is  greatest  over  the  arcs  ^N',  <?'N,  which  con- 
tain the  syzigies,  (649,)  and  as  these  arcs  are  also  longer  than  the  arcs  Ny, 
N'^',  the  node  will,  on  the  whole,  retrograde  each  revolution.     The  velocity 
of  retrogradation  will,  however,  be  less  than  when  the  nodes  are  in  quadra- 
tures, and  proportionably  less  as  the  distance  of  the  sun  from  this  position  is 
greater. 

In  the  position  represented  in  Fig.  121, 
a  direct  motion  will  take  place  over  the 
arcs  ^'N'  and  q~N ;  but  as  Ny'  and  N'y,  the 
arcs  of  retrograde  motion,  are  of  greater 
extent  than  ^'N'  and  j-N,  and  moreover 
contain  the  syzigies,  the  retrograde  motion 
in  each  revolution  must  exceed  the  direct, 
as  before. 

If  we  suppose  the  sun  to  be  situated  on 
the  other  side  of  the  line  of  nodes,  the 
effect  of  the  disturbing  force  will  obviously 
be  the  same  in  any  one  position  of  the  sun, 
as  in  the  position  diametrically  opposite  to 
it.  It  appears,  then,  that  the  line  of  the 
nodes  has  a  retrograde  motion  in  every 
possible  position  of  the  sun. 

653.  We  have  thus  far  supposed  the  sun 
to  remain  stationary  in  the  various   posi- 
tions in  which  we  have  supposed  it,  during  the  revolution  of  the  moon.     It 
remains,  then,  to  consider  the  effect  of  the  sun's  motion  in  this  interval.  And 
first,  it  is  plain,  that,  as  the  sun  advances  from  S  towards  N',  (Fig.  120,)  the 
arcs  Ny,  NV  will  increase,  and  the  arcs  <?N'  and  <?'N  diminish ;  from  which 
it  appears,  that,  during  the  advance  of  the  sun  from  the  point  90°  behind  the 
descending  node  to  this  node,  its  motion  in  the  course  of  each  revolution  of 
the  moon  will  cause  the  retrograde  motion  of  the  node  to  be  slower  than  it 
otherwise  would  be.     While  the  sun  moves  from  the  ascending  node  to  the 
90°  from  it,  the  effect  of  its  motion  will  obviously  be  just  the  reverse  of  this. 
During  its  passage  from  the  descending  to  the  ascending  node,  the  effect  will 
be  the  same  in  either  quadrant  as  in  that  diametrically  opposite. 

The  variation  in  the  intensity  of  the  perpendicular  force  conspires  with 
the  difference  of  situation  of  the  sun  and  its  motion  during  a  revolution  of  the 
moon  in  diminishing  or  increasing,  as  the  case  may  be,  the  velocity  of  retro- 
gradation  of  the  nodes. 

654.  Let  us  now  treat  of  the  change  of  the  inclination  of  the  orbit,  result 


248         PERTURBATIONS  OF  ELLIPTIC  MOTION  OF  THE  MOON. 


ing  from  the  disturbing  action  of  the  sun.  And  first,  if  we  refer  to  Fig.  120 
we  shall  see  that  when  the  nodes  are  in  quadrature  the  inclination  will  dimin- 
ish while  the  moon  is  moving  from  the  ascending  node  N  to  the  point  L  90° 
distant  from  it,  and  increase  while  she  is  moving  from  L  to  the  other  node 
N'.  In  the  other  half  of  the  orbit  the  tendency  of  the  disturbing  force  is  the 
same,  (650  ;)  and  therefore  while  the  moon  is  moving  from  N'  to  L'  the  in- 
clination will  diminish,  and  while  she  is  moving  from  L'  to  N  it  will  increase. 
The  diminutions  and  increments  will  compensate  each  other,  and  the  original 
inclination  will  be  regained  at  the  close  of  the  revolution. 

When  the  nodes  are  in  syzigies  there  will  be  no  change  of  inclination, 
(649.) 

655.  In  the  situations  of  the  sun  represented  in  Figs.  120  and  121  the 
inclination  will  decrease  from  q  to  L  and  from  q'  to  L',  and  increase  from  L 
to  q'  and  from  L'  to  q,  the  effects  being  the  same  as  when  the  nodes  are  in 
quadratures  over  the  arcs  qL  and  LN'  in  Fig.  120,  and  NL  and  Lq'  in  Fig. 


Fig.  120. 


Fig.  121. 


121,  and  being  reversed  over  the  arcs  Ng  and  Ny  in  Fig.  120,  and  <?N  and 
y'N'  in  Fig.  121.  When  the  sun  has  the  position  represented  in  Fig.  120, 
the  arcs  of  increase  Lq'  and  L'q  will  be  greater  than  the  arcs  of  diminution 
qL  and  q'L'.  The  disturbing  force  will  also  be  greater  in  the  former  arcs 
than  in  the  latter.  In  the  position  supposed,  therefore,  there  will  be,  on  the 
whole,  an  increase  of  inclination  eVery  revolution.  When  the  sun  is  in  the 
position  represented  in  Fig.  121,  the  arcs  of  diminution  qL  and  q'L  will  be 
the  greater  ;  and  the  force  in  them  will  also  be  the  greater.  In  this  case, 
therefore,  there  will  be  a  diminution  of  the  inclination  each  revolution  of  the 
moon. 

When  the  sun  is  on  the  other  side  of  the  line  of  nodes,  the  results  will  be 
the  same  as  in  the  positions  diametrically  opposite. 

656.  To  inquire  now  into  the  consequences  of  the  sun's  motion  during  the 
revolution  of  the  moon.     As  the  sun  moves  from  S  towards  N'  (Fig.  120) 
the  arcs  Lq',  L'q,  over  which  there  is  an  increase  of  the  inclination,  will  in- 
crease ;  and  the  arcs  qL,  q'L',  over  which  there  is  a  diminution,  will  diminish. 
The  motion  of  the  sun  will,  therefore,  in  approaching  the  descending  node, 
render  the  increase  of  the  inclination  each  revolution  of  the  moon  greater  than 
it  otherwise  would  be.     When  the  sun  is  receding  from  the  ascending  node, 
the  corresponding  arcs  will  experience  corresponding  changes,  and  therefore 
the  diminution  will  now  be  less  than  if  the  sun  were  stationary. 

The  results  will  be  similar  for  the  opposite  quadrants  on  the  other  side  of 
the  line  of  nodes. 

657.  Since  the  inclination  diminishes  as  the  sun  recedes  from  either  node, 


PLANETARY  PERTURBATIONS.  249 

and  increases  as  it  approaches  either  node,  it  will  be  the  least  when  the  nodes 
are  in  quadratures,  and  the  greatest  when  they  are  in  syzigies. 

It  is  important  to  observe  that  the  change  of  inclination  which  we  have 
been  considering  is  modified  by  the  retrograde  motion  of  the  node  ;  and  thus, 
that,  besides  the  variations  of  this  element  connected  with  the  motions  of  the 
moon  and  sun,  there  is  another  extending  through  the  period  employed  by  the 
node  in  completing  a  revolution  with  respect  to  both  the  sun  and  moon. 

658.  The  perturbations  of  the  elliptic  motion  of  the  moon,  comprising  ine- 
qualities of  orbit  longitude,  and  variations  in  the  form  and  position  of  the  orbit, 
which  have  now  been  under  consideration,  depend  upon  the  configurations  of 
the  sun  and  moon,  with  respect  to  each  other,  the  perigee  of  each  orbit,  and 
the  node  of  the  moon's  orbit.     Their  effects  will  disappear  when  the  configu- 
rations upon  which  they  depend  become  the  same.     They  are  therefore  pe- 
riodical. 

659.  The  perturbations  of  the  motions  of  a  planet,  produced  by  the  action 
of  another  planet,  are  precisely  analogous  to  the  perturbations  of  the  motions 
of  the  moon,  produced  by  the  action  of  the  sun.     The  disturbing  forces  are 
obviously  of  the  same  kind,  and  they  are  subject  to  variations  from  precisely 
similar  causes.     But,  owing  to  the  smallness  of  the  masses  of  the  planets  and 
their  great  distances,  their  disturbing  forces  are  much  more  minute  than  the 
disturbing  force  of  the  sun.     From,  this  cause,  together  with  the  slow  rela- 
tive motion  of  the  disturbing  and  disturbed  body,  the  motion  of  the  apsides  and 
nodes,  and  the  accompanying  variations  of  eccentricity  and  inclination,  are 
very  much  more  gradual  in  the  case  of  the  planets  than  in  the  case  of  the 
moon.    Their  periods  comprise  many  thousands  of  years,  and  on  this  account 
they  are  called  Secular  Motions  or  Variations.    In  consequence  of  the  greater 
feebleness  of  the  disturbing  forces,  the  periodical  inequalities  are  also  much 
less  in  amount.     Moreover,  as  the  motion  of  a  planet  is  much  slower  than 
that  of  the  moon,  and  as  the  variations  of  its  orbit  are  more  gradual  than 
those  of  the  lunar  orbit,  the  compensations  produced  by  a  change  of  configu- 
rations are  much  more  slowly  effected,  and  thus  the  periods  of  the  inequali- 
ties are  much  longer. 

660.  The  motions  of  the  moon  would  be  subject  to  no  secular  variations  if 
the  apparent  orbit  of  the  sun  were  unchangeable  ;  but  the  secular  variation 
of  the  eccentricity  of  the  sun's  orbit,  which  answers  to  an  equal  variation  of 
the  eccentricity  of  the  earth's  orbit,  that  is  produced  by  the  action  of  the 
planets,  gives  rise  to  a  secular  inequality  in  the  motion  of  the  moon,  called 
the  Acceleration  of  the  Moon.    This  inequality  was  discovered  from  observa- 
tion.    Its  physical  cause  was  first  made  known  by  Laplace. 


CHAPTER  XXII. 

OF  THE  RELATIVE  MASSES  AND  DENSITIES  OP  THE  SUN,  MOON,  AND 
PLANETS  j  AND  OF  THE  RELATIVE  INTENSITY  OF  THE  GRAVITY 
AT  THEIR  SURFACE. 

661.  THE  perturbations  which  a  planet  produces  in  the  motions 
of  the  other  planets,  depend  for  their  amount  chiefly  upon  the  ra- 
tio of  the  mass  of  the  planet  to  the  mass  of  the  sun,  and  the  ratio 
of  the  distance  of  the  planet  from  the  sun  to  the  distance  of  the 
planet  disturbed  from  the  same  body.  Now,  the  ratio  of  the  dis- 

32 


250    RELATIVE  MASSES  OF  THE  SUN,  MOON,  AND  PLANETS. 

tances  is  known  by  the  methods  of  Plane  Astronomy ;  conse- 
quently, the  observed  amount  of  the  perturbations  ought  to  make 
known  the  ratio  of  the  masses,  the  only  unknown  element  upon 
which  it  depends: 

This  is  one  method  of  determining  the  masses  of  the  planets. 
The  masses  of  those  planets  which  have  satellites  may  be  found 
by  another  and  simpler  method,  viz. :  by  comparing  the  attractive 
force  of  the  planet  for  either  one  of  its  satellites  with  the  attract- 
ive force  of  the  sun  for  the  planet.  These  forces  are  to  each  other 
directly  as  the  masses  of  the  planet  and  sun,  and  inversely  as  the 
squares  of  the  distances  of  the  satellite  from  the  primary  and  of 
the  primary  from  the  sun.  Thus,  calling  the  forces  j,  F,  the 
masses  m,  M,  and  the  distances  d,  D,  we  have 

f   F      m    M 

/  •        •  #  -  D2  » 

whence  we  obtain  m  :  M  :  :  fd?  :  FD2.  If  we  regard  the  orbits  as 
circles,  then  d  and  D  will  be  the  mean  distances,  respectively,  of 
the  satellite  from  the  primary,  and  of  the  primary  from  the  sun, 
and  are  given  in  tables  II,  III,  and  VI.  The  ratio  of  /  to  F  is 
equal  to  the  ratio  of  the  versed  sines  of  the  arcs  described  by  the 
satellite  and  primary,  in  some  short  interval  of  time  ;*  since  these 
are  sensibly  equal  to  the  distances  that  the  two  bodies  are  deflect- 
ed in  this  interval  from  the  tangents  to  their  orbits,  towards  the 
centres  about  which  they  are  revolving :  and  since  the  rates  of 
motion  and  dimensions  of  the  orbits  of  the  planet  and  satellites  are 
known,  these  arcs  and  their  versed  sines  are  easily  determined. 

662.  The  second  column  of  Table  IV  exhibits  the  relative 
masses  of  the  sun,  moon,  and  planets,  according  to  the  most  re- 
ceived determinations,  that  of  the  sun  being  denoted  by  1 . 

663.  The  quantities  of  matter  of  the  sun,  moon,  and  planets,  as 
well  as  their  bulks,  being  known,  their  densities  may  be  easily 
computed ;  for,  the  densities  of  bodies  are  proportional  to  their 
quantities  of  matter  divided  by  their  bulks.     The  third  column  of 
Table  IV  contains  the  densities  of  the  sun,  moon,  and  planets,  that 
of  the  earth  being  denoted  by  1 .     It  will  be  seen  on  inspecting  it, 
that,  for  the  most  part,  the  densities  of  the  planets  decrease  as  we 
recede  from  the  sun. 

664.  The  relative  intensity  of  the  gravity  at  the  surface  of  the 
sun,  moon,  and  planets,  may  also  readily  be  found,  when  the 
masses  and  bulks  of  these  bodies  are  known.      For  supposing 
them  to  be  spherical,  and  not  to  rotate  on  their  axes,  the  gravity 
at  their  surface  will  be  directly  as  their  masses  and  inversely  as 
the  squares  of  their  radii,  or,  in  other  words,  proportional  to  their 
masses  divided  by  the  squares  of  their  radii.     The  centrifugal 
force  at  the  surface  of  a  planet,  generated  by  its  rotation  on  its 

*  It  is  to  be  observed  that  the  versed  sines  here  mentioned  relate  to  the  actual 
arcs  described  in  the  two  unequal  orbits. 


EXPLANATION  OF  SPHEROIDAL  FORM  OF  THE  EARTH.          251 

axis,  diminishes  the  gravity  due  to  the  attraction  of  the  matter  of 
the  planet.  The  diminution  thus  produced  on  any  of  the  planets 
is  not,  however,  very  considerable.  The  method  of  determining 
the  centrifugal  force  at  the  surface  of  a  body  in  rotation,  is  given  in 
treatises  on  Mechanics.  (See  Courtenay's  Mechanics,  pages  250 
and  251.) 

The  fourth  column  of  Table  IV  exhibits  the  relative  intensity 
of  the  gravity  at  the  surface  of  the  sun,  moon,  and  planets,  that  at 
the  surface  of  the  earth  being  denoted  by  1 . 


CHAPTER    XXIII. 

OP  THE  FIGURE  AND  ROTATION  OF  THE  EARTH  ;  AND  OF  THE  PRE- 
CESSION OF  THE  EQUINOXES  AND  NUTATION. 

665.  WE  have  already  seen  (159)  that  measurements  made  upon 
the  earth's  surface  establish  that  the  figure  of  the  earth  is  that  of 
an  oblate  spheroid,  and  that  the  oblateness  at  the  poles  is  about  ¥£j. 

666.  From  the  amount  and  law  of  the  variation  of  the  force  of 
gravity  upon  the  earth's  surface,  ascertained  by  observations  upon 
the  length  of  the  seconds'  pendulum,  it  is  proved  that  the  matter 
of  the  earth  is  not  homogeneous,  but  denser  towards  the  centre, 
and  that  it  is  arranged  in  concentric  strata  of  nearly  an  elliptical 
form  and  uniform  density. 

The  fact  of  the  greater  density  of  the  earth  towards  its  centre 
has  also  been  established  by  observations  upon  the  deviation  of  a 
plumb-line  from  the  vertical,  produced  by  the  attraction  of  a  moun- 
tain ; — the  amount  of  the  deviation  being  ascertained  by  observing 
the  difference  in  the  zenith  distance  of  the  same  star,  as  measured 
with  a  zenith-sector  on  opposite  sides  of  the  mountain.  To  the 
north  of  the  mountain  the  plummet  was  drawn  towards  the  south 
and  the  zenith  distance  of  a  star  to  the  north  of  the  zenith  was 
diminished  ;  while  to  the  south  of  the  mountain  the  plummet  was 
drawn  towards  the  north,  and  the  zenith  distance  of  the  same  star 
was  increased  by  an  equal  amount :  and  thus  the  difference  of  the 
two  measured  zenith  distances  was  equal  to  twice  the  deviation  of 
the  plumb-line  from  the  true  vertical  in  either  of  the  positions  of 
the  instrument ;  (allowance  being  made  for  the  difference  of  lati- 
tude of  the  two  stations,  as  determined  from  the  distance  between 
them  and  the  known  length  of  a  degree.) 

Such  observations  were  made  for  the  purpose  of  determining  the 
mean  density  of  the  earth  by  Dr.  Maskelyne,  in  1774,  on  the  sides 
of  the  mountain  Schehallien  in  Scotland.  The  observed  deviation 
of  the  plumb-line  made  known  the  ratio  of  the  attraction  of  the 
mountain  to  that  of  the  whole  earth,  and  thus  the  relative  quanti- 
ties of  matter  in  the  mountain  and  earth.  These  being  ascertained 


252         OF  THE  FIGURE  AND  ROTATION  OF  THE  EARTH,  ETC. 

and  the  figure  and  bulk  of  the  mountain  having  been  determined 
by  a  survey,  the  relative  density  of  the  earth  and  mountain  became 
known  by  the  principle  mentioned  in  Art.  663,  and  thence  the  ac- 
tual density  of  the  earth,  the  density  of  the  mountain  having  been 
found  by  experiment.  The  result  was,  that  the  mean  density  of 
the  earth  is  4.95,  the  density  of  water  being  1 . 

667.  The  spheroidal  form  of  the  surface  of  the  earth  and  of  its 
internal  strata  is  easily  accounted  for,  if  we  suppose  the  earth  to 
have  been  originally  in  a  fluid  state.     The  tendency  of  the  mutual 
attraction  of  its  particles  would  be  to  give  it  a  spherical  form  ;  but 
by  virtue  of  its  rotation,  all  its  particles,  except  those  lying  imme- 
diately on  the  axis,  would  be  animated  by  a  centrifugal  force  in- 
creasing with  their  distance  from  the  axis.     If,  therefore,  we  con- 
ceive of  two  columns  of  fluid  extending  to  the  earth's  centre,  one 
from  near  the  equator,  and  the  other  from  near  either  pole,  the 
weight  of  the  former  would  by  reason  of  the  centrifugal  force  be 
less  than  that  of  the  latter.     In  order,  then,  that  they  may  sustain 
each  other  in  equilibrio,  that  near  the  equator  must  increase  in 
length,  and  that  near  the  pole  diminish.     As  this  would  be  true  at 
the  same  time  for  every  pair  of  columns  situated  as  we  have  sup- 
posed, the  surface  of  the  whole  body  of  fluid  about  the  poles  must 
fall,  and  that  of  the  fluid  about  the  equator  rise.     In  this  manner 
the  earth  would  become  flattened  at  the  poles  and  protuberant  at 
the  equator. 

668.  Upon  a  strict  investigation  it  appears  that  a  homogeneous 
fluid  of  the  same  mean  density  with  the  earth,  and  rotating  on  its 
axis  at  the  same  rate  that  the  earth  does,  would  be  in  equilibrium, 
if  it  had  the  figure  of  an  oblate  spheroid,  of  which  the  axis  was 
to  the  equatorial  diameter  as  229  to  230,  or  of  which  the  oblate- 
ness  was  ¥|  o  •     If  the  fluid  mass  supposed  to  rotate  on  its  axis  be 
not  homogeneous,  but  be  composed  of  strata  that  increase  in  den- 
sity from  the  surface  to  the  centre,  the  solid  of  equilibrium  will 
still  be  an  elliptic  spheroid,  but  the  oblateness  will  be  less  than 
when  the  fluid  is  homogeneous. 

669.  The  time  of  the  earth's  rotation,  as  well  as  the  position  of 
its  axis,  would  change  if  any  variation  should  take  place  in  the 
distribution  of  the  matter  of  the  earth,  or  in  case  of  the  impact  of 
a  foreign  body. 

If  any  portion  of  matter  be,  from  any  cause,  made  to  approach 
the  axis,  its  velocity  will  be  diminished,  and  the  velocity  lost  being 
imparted  to  the  mass,  will  tend  to  accelerate  the  rotation.  If  any 
oortion  of  matter  be  made  to  recede  from  the  axis,  the  opposite 
effect  will  be  produced,  or  the  rotation  will  be  retarded.  In  point 
of  fact,  the  changes  that  take  place  in  the  position  of  the  matter 
of  the  earth,  whether  from  the  washing  of  rains  upon  the  sides  of 
mountains,  or  evaporation,  or  any  other  known  cause,  are  not  suf- 
ficient ever  to  produce  any  sensible  alteration  in  the  circumstances 
of  the  earth's  rotation  on  its  axis. 


PHYSICAL  THEORY  OF  PRECESSION  AND  NUTATION.  25.1 

670.  It  is  ascertained  from  direct  observation,  that  there  has  in 
reality  been  no  perceptible  change  in  the  period  of  the  earth's  ro 
tation  since  the  time  of  Hipparchus,  120  years  before  the  begin- 
ning of  the  present  era.     We  may  therefore  conclude,  a  posteriori, 
that  there  has  been  no  material  change  in  the  form  and  dimensions 
of  the  earth  in  this  interval. 

671.  Were  the  axis  of  the  earth  to  experience  any  change  of 
position  with  respect  to  the  matter  of  the  earth,  the  latitudes  of 
places  would  be  altered.     A  motion  of  200  feet  might  increase  or 
diminish  the  latitude  of  a  place  to  the  amount  of  2",  an  angle  which 
can  be  measured  by  modern  instruments.     Now,  in  point  of  fact, 
the  latitudes  of  places  have  not  sensibly  varied  since  their  first  de- 
termination with  accurate  instruments  ;  therefore,  in  this  interval 
the  axis  of  the  earth  cannot  have  materially  changed.     Indeed, 
since  the  earth's  surface  and  its  internal  strata  are  arranged  sym- 
metrically with  respect  to  the  present  axis  of  rotation,  it  is  to  be  in- 
ferred that  this  axis  is  the  same  as  that  which  obtained  at  the  epoch 
when  the  matter  of  the  earth  changed  from  a  fluid  to  a  solid  state. 

672.  The  motions  of  the  earth's  axis,  along  with  the  whole  body 
of  the  earth,  which  give  rise  to  the  Precession  of  the  Equinoxes 
and  Nutation,   are  consequences   of  the  spheroidal  form  of  the 
earth,  inasmuch  as  they  are  produced  by  the  actions  of  the  sun 
and  moon  upon  that  portion  of  the  matter  of  the  earth  which  lies 
on  the  outside  of  a  sphere  conceived  to  be  described  about  the 
earth's  axis.     The  physical  theory  of  the  phenomena  in  question 
is  analogous  to  that  of  the  retrogradation  of  the  moon's  nodes.  The 
sun  produces  a  retrograde  movement  of  the  points  in  which  the 
circle  described  by  each  particle  of  the  protuberant  mass  cuts  the 
plane  of  the  ecliptic,  as  it  does  of  the  moon's  nodes  ;  the  effect 
produced  is,  however,  exceedingly  small,  by  reason  of  the  inertia 
of  the  interior  spherical  mass  connected  with  the  external  mass 
upon  which  the  action  takes  place.     The  moon,  in  like  manner, 
occasions  a  retrograde  movement  of  the  nodes  of  the  same  parti- 
cles on  the  plane  of  its  orbit.     The  actions  of  the  sun  and  moon 
will  not  be  the  same  each  revolution  of  a  particle.     That  of  the 
sun  will  vary  during  the  year  with  the  angular  distance  of  the  sun 
from  the  node,  (649  ;)  and  that  of  the  moon  will  vary  during  each 
month  with  the  distance  of  the  moon  from  the  node,  and  also 
during  a  revolution  of  the  nodes  of  the  moon's  orbit  by  reason  of 
the  change  in  the  inclination  of  the  orbit  to  the  equator.     The 
mean  effect  of  both  bodies  is  the  precession;  the  inequality  re- 
sulting from  the  change  in  the  sun's  action  during  the  year  is  the 
solar  nutation ;  and  the  inequality  consequent  upon  the  retrogra- 
dation  of  the  moon's  nodes  is  the  lunar  nutation,  or  the  chief 
part  of  it :  the  change  in  the  position  of  the  equinox  occasioned  by 
the  moon's  revolution,  never  exceeds  \  of  a  second  of  an  arc;  and  the 
change  of  the  obliquity  of  the  ecliptic  from  this  cause  is  still  less- 


254  OP  THE  TIDES. 

CHAPTER    XXIV. 

OF  THE  TIDES. 

673.  THE  alternate  rise  and  fall  of  the  surface  of  the  ocean 
twice  in  the  course  of  a  lunar  day,  or  about  25  hours,  is  the  phe- 
nomenon known  by  the  name  of  the  Tides.    The  rise  of  the  water 
is  called  the  Flood  Tide,  and  the  fall  the  Ebb  Tide. 

674.  The  interval  between  one  high  water  and  the  next  is,  at  a 
mean,  half  a  mean  lunar  day,  or  12h.  25m.  14s.     Low  water  has 
place  nearly,  but  not  exactly,  at  the  middle  of  this  interval ;  the 
tide,  in  general,  employing  nine  or  ten  minutes  more  in  ebbing  than 
in  flowing.     As  the  interval  between  one  period  of  high  water  and 
the  second  following  one  is  a  lunar  day,  or  Id.  Oh.  50m.  28s.,  the 
retardation  in  the  time  of  high  water  from  one  day  to  another  is 
50m.  28s.,  in  its  mean  state. 

675.  The  time  of  high  water  is  mainly  dependent  upon  the  po- 
sition of  the  moon,  being  always,  at  any  given  place,  about  the 
same  length  of  time  after  the  moon's  passage  over  the  superior  or 
inferior  meridian.     As  to  the  length  of  the  interval  between  the 
two  periods,  at  different  places,  in  the  open  sea  it  is  only  from  two 
to  three  hours ;  but  on  the  shores  of  continents,  and  in  rivers, 
where  the  water  meets  with  obstructions,  it  is  very  different  at 
different  places,  and  in  some  instances  is  of  such  length  that  the 
time  of  high  water  seems  to  precede  the  moon's  passage. 

676.  The  height  of  the  tide  at  high  water  is  not  always  the 
same,  but  varies  from  day  to  day ;  and  these  variations  have  an 
evident  relation  to  the  phases  of  the  moon.     It  is  greatest  at  the 
syzigies  ;  after  which  it  diminishes  and  becomes  the  least  at  the 
quadratures.* 

677.  The  tides  which  occur  near  the  syzigies,  are  called  the 
Spring  Tides ;    and  those  which  occur  near  the  quadratures  are 
called  the  Neap  Tides. 

The  highest  of  the  spring  tides  is  not  that  which  has  place 
nearest  to  new  or  full  moon,  but  is  in  general  the  third  following 
tide.  In  like  manner  the  lowest  of  the  neap  tides  is  the  third  or 
fourth  tide  after  the  quadrature. 

The  spring  tides  are,  in  general,  about  twice  the  height  of  the 
neap  tides.  At  Brest,  in  France,  the  former  rises  to  the  height  of 
19.3  feet,  and  the  latter  only  to  9.2  feet.  In  the  Pacific  Ocean  the 
highest  of  the  tides  of  the  syzigies  is  5  feet,  and  the  lowest  of  the 
tides  of  the  quadratures  is  between  2  and  2.5  feet. 

678.  The  tides  are  also  affected  by  the  declinations  of  the  sun 
and  moon :  thus,  the  highest  spring  tides  in  the  course  of  the  year 

*  Sally's  Astronomical  Tables  and  Formulae,  p.  25. 


'* 

PHENOMENA  OF  THE  TIDES.  <  >>     255   ^     < 

ft        /"  -/' 

are  those  which  occur  near  the  equinoxes.  The  djjctraor^narily  hi^h  /  . 
tides  which  frequently  occur  at  the  equinoxes  are,  ho^ypr,  in      f ' 
part  attributable  to  the  equinoctial  gales.    Also,  when  the  moda'or, 
the  sun  is  out  of  the  equator,  the  evening  and  morning  tides  diflEw,.\* 
somewhat  in  height.    At  Brest,  in  the  syzigies  of  the  summer  sol- 
stice, the  tides  of  the  morning  of  the  first  and  second  day  after  the 
syzigy  are  smaller  than  those  of  the  evening  by  6.6  inches.    They 
are  greater  by  the  same  quantity  in  the  syzigies  of  the  winter  sol- 
stice.* 

679.  The  distance  of  the  moon  from  the  earth  has  also  a  sensi- 
ble influence  upon  the  tides.    In  general,  they  increase  and  dimin- 
ish as  the  distance  increases  and  diminishes,  but  in  a  more  rapid 
ratio. 

680.  The  daily  retardation  of  the  time  of  high  water  varies  with 
the  phases  of  the  moon.  It  is  at  its  minimum  towards  the  syzigies, 
when  the  tides  are  at  their  maximum ;  and  it  is  then  about  40m. 
But,  towards  the  quadratures,  when  the  tides  are  at  their  minimum, 
the  retardation  is  the  greatest  possible  ;  and  amounts  to  about  Ih. 
15m. 

The  variation  in  the  distance  of  the  sun  and  moon  from  the  earth, 
(and  particularly  the  moon,)  has  an  influence  also  on  this  retarda- 
tion. 

The  daily  retardation  of  the  tides  varies  likewise  with  the  decli- 
nation of  the  sun  and  moon.t 

681.  The  facts  which  have  been  detailed  indicate  that  the  tides 
are  produced  by  the  actions  of  the  sun  and  moon  upon  the  waters 
of  the  ocean ;  but  in  a  greater  degree  by  the  action  of  the  moon. 
To  explain  them,  let  us  suppose  at  first  that  the  whole  surface  of 
the  earth  is  covered  with  water.     We  remark,  in  the  first  place, 
that  it  is  not  the  whole  attractive  force  of  the  moon  or  sun  which 
is^effective  in  raising  the  waters  of  the  ocean,  but  the  difference  in 
the  actions  of  each  body  upon  the  different  parts  of  the  earth  ;  or, 
more  precisely,  that  the  phenomenon  of  the  tides  is  a  consequence 
of  the  inequality  and  non-parallelism  of  the  attractive  forces  exert- 
ed by  the  moon,  as  well  as  by  the  sun,  upon  the  different  particles 
of  the  earth's  mass.     From  this  cause  there  results  a  diminution 
in  the  gravity  of  the  particles  of  water  at  the  surface,  for  a  certain 
distance  about  the  point  immediately  under  the  moon,  and  the  point 
diametrically  opposite  to  this,  and  an  augmentation  for  a  certain 
distance  on  the  one  side  and  the  other  of  the  circle  90°  distant  from 
these  points,  or  of  which  they  are  the  geometrical  poles  :  in  con- 
sequence of  which  the  water  falls  about  this  circle  and  rises  about 
these  points.     That  the  actions  of  the  moon  upon  the  different 
parts  of  the  earth's  mass  are  really  unequal  is  evident,  from  the 
fact,  that  these  parts  are  at  different  distances  from  the  moon.    To 


*  Laplace's  System  of  the  World.         t  Daily's  Tables  and  Formal®,  p.  26. 


256 


OF  THE  TIDES. 


show  that  the  inequality  will  give  rise  to  the  results  just  noted,  let 
us  suppose  that  the  circle  acbd  (Fig.  122)  represents  the  earth,  and 
M  the  place  of  the  moon ;  then  a  will  be  the  point  of  the  earth's 

surface  directly  under  the  moon,  b  the 
point  diametrically  opposite  to  this,  and 
the  right  line  dc  perpendicular  to  MO 
will  represent  the  circle  traced  on  the 
earth's  surface  90°  distant  from  a  and  b. 
Now,  the  attraction  of  the  moon  for  the 
general  mass  of  the  earth  is  the  same  as 
if  the  whole  mass  were  concentrated  at 
the  centre  O.  But  the  centre  of  the 
earth  is  more  distant  from  the  moon 
than  the  point  a  at  the  surface.  It  fol- 
lows, therefore,  that  a  particle  of  matter 
situated  at  a  will  be  drawn  towards  the 
moon  with  a  proportionally  greater  force 
than  the  centre,  or  than  the  general  mass 
of  the  earth.  Its  gravity  or  tendency 
towards  the  earth's  centre  will  therefore 
be  diminished  by  the  amount  of  this  ex- 
cess. On  the  other  hand,  the  centre  is 
nearer  to  the  moon  than  the  point  b.  It 
is  therefore  attracted  more  strongly  than 
a  particle  at  b.  The  excess  will  be  a 
force  tending  to  draw  the  centre  away 
from  the  particle ;  and  the  effect  will 
be  the  same  as  if  the  particle  were  drawn  away  from  the 
centre  by  the  same  force  acting  in  the  opposite  direction.  The 
result  then  is,  that  this  particle  has  its  gravity  towards  the  earth's 
centre  diminished,  as  well  as  the  particle  at  a.  If  now  we  consider 
a  particle  at  some  point  t  near  to  a,  the  moon's  action  upon 
it  (tr)  may  be  considered  as  taking  effect  partially  in  the  direction 
tk  parallel  to  OM,  and  partially  in  the  direction  of  the  tangent  or 
horizontal  line  ts.  The  component  (ts)  in  the  latter  direction,  will 
have  no  tendency  to  alter  the  gravity  of  the  particle  towards  the 
earth's  centre.  The  component  (sr)  in  the  direction  tk,  will  obvi- 
ously be  less  than  the  actual  force  of  attraction  tr ;  and  the  dif- 
ference will  be  greater  in  proportion  as  the  particle  is  more  remote 
from  a.  But  this  component  will  decrease  gradually  from  a,  while 
the  attraction  for  the  centre  is  less  than  for  a  by  a  certain  finite  differ- 
ence :  it  is  plain,  therefore,  that  the  component  in  question  will  be 
greater  than  the  attraction  for  the  centre,  in  the  vicinity  of  the  point 
a,  and  for  a  certain  distance  from  it  in  all  directions.  The  gravity 
of  the  particles  will  therefore  be  diminished  for  a  certain  distance 
from  this  point.  In  a  similar  manner  it  may  be  shown  that  it  will 
also  be  diminished  for  a  certain  distance  from  the  point  b.  Let  us 
now  consider  a  particle  at  c,  90°  from  the  points  a  and  b.  The  at- 


PHYSICAL  THEORY  OF  THE  TIDES.  257 

traction  of  the  moon  for  it  will  take  effect  in  the  two  directions  cl 
and  cO.  The  force  in  the  latter  direction  alone  will  alter  the  grav- 
ity of  the  particle  ;  and  this,  it  is  plain,  will  increase  it.  The  same 
effect  will  extend  to  a  certain  distance  from  c  in  both  directions. 

A  strict  mathematical  investigation  would  show  that  the  gravity 
is  diminished  for  a  distance  of  55°  from  a  and  b  in  all  directions ; 
and  is  augmented  for  a  distance  of  35°  on  each  side  of  the  circle 
dc,  90°  distant  from  the  points  a  and  b.  These  distances  are  rep- 
resented in  the  Figure. 

This  may  be  easily  made  out  by  means  of  the  expression  for  the  radial  disturb, 
ing  force  of  the  sun  in  its  action  upon  the  moon,  (643,)  viz.  —  y  (1  — 3  cos*  0).  If 

we  consider  m  as  denoting  the  mass  of  the  moon,  a  the  moon's  distance  from  the 
earth's  centre,  y  the  distance  of  a  particle  of  matter  at  some  point  t  of  the  earth's 
surface  from  the  earth's  centre,  and  0  the  angular  distance  or  elongation  (MO<) 
of  the  same  particle  from  the  moon,  as  seen  from  the  centre  of  the  earth,  it  will  ex- 
press the  change  in  the  gravity  of  a  particle  at  the  earth's  surface,  produced  by  the 
moon's  action.  The  points  a  and  b  will  answer  to  conjunction  and  opposition,  and 
Ihe  points  c  and  d  to  the  quadratures.  Now  we  have  already  seen  (643)  that  the 
gravity  of  the  moon  is  increased  at  the  quadratures,  and  for  35°  on  each  side  of 
them  ;  and  diminished  at  the  syzigies,  and  55°  from  them  in  both  directions.  It  fol- 
lows, therefore,  that  the  same  is  true  for  particles  of  matter  at  the  earth's  surface. 

In  consequence  of  the  earth's  diurnal  rotation,  the  parts  of  the 
surface,  at  which  the  rise  and  fall  of  the  water  will  take  place,  will 
be  continually  changing.  Were  the  entire  rise  and  fall  produced 
instantaneously,  the  points  of  highest  water  would  constantly  be  the 
precise  points  in  which  the  line  of  the  centres  of  the  moon  and 
earth  intersects  the  surface,  and  it  would  always  be  high  water  on 
the  meridian  passing  through  these  points,  both  in  the  hemisphere 
where  the  moon  is,  and  in  the  opposite  one.  On  the  west  side  of 
this  meridian,  the  tide  would  be  flowing ;  on  the  east  side  of  it,  it 
would  be  ebbing ;  and  on  the  meridian  at  right  angles  to  the  same, 
it  would  be  low  water.  Bat  it  is  plain  that  the  effects  of  the  moon's 
action  will  not  be  instantaneously  produced,  and  therefore  that  the 
points  of  highest  water  will  fall  behind  the  moon.  It  appears  from 
observation,  that  in  the  open  sea  the  meridian  of  high  water  is  about 
30°  to  the  east  of  the  moon. 

The  great  tide  wave  thus  raised  by  the  moon,  and  which  follows 
it  in  its  diurnal  motion,  will  be  a  mere  undulation,  or  alternate  rise 
and  fall  of  the  water,  without  any  progressive  motion,  if,  as  we  have 
supposed,  it  is  nowhere  obstructed  by  shallows,  islands,  or  the 
shores  of  continents. 

682.  It  is  evident  that  the  sun  will  produce  precisely  similar 
effects  with  the  moon,  and  will  raise  a  tide  wave  similar  to  the 
lunar  tide  wave,  which  will  follow  it  in  its  diurnal  motion. 

683.  To  show  that  the  effects  of  the  sun  are  less  in  degree  than  those  of  the 
moon,  let  us  take  the  general  expression  for  the  change  of  the  moon's  gravity, 
arising  from  the  action  of  the  sun,  namely, 

...(a), 
33 


258  OF  THE  TIDES 

in  which  -m  denotes  the  mass  of  the  sun,  a  its  distance,  (the  mean  distance  of  the 
moon  being  taken  as  1,)  y  the  distance  of  the  moon  in  its  given  position,  and  0  its 
elongation  from  the  sun,  as  seen  from  the  earth's  centre.  This  formula  will  serve 
to  express  the  change*  in  the  gravity  of  a  particle  of  matter  upon  the  earth's  sur- 
face, produced  by  the  sun's  action,  if  we  take  m  =  the  mass  of  the  sun-,  as  before, 
a  =  its  distance  expressed  in  terms  of  the  radius  of  the  earth  as  unity,  y  =  the 
distance  of  the  particle  from  the  centre  of  the  earth,  and  <f>  =  its  elongation  from 
the  sun,  as  seen  from  the  earth's  centre.  If  we  designate  the  corresponding  quan- 
tities for  the  moon  by  m',  a',  y,  <f>,  we  shall  have  for  the  change  of  the  gravity  of 
a  particle,  produced  by  the  moon's  action, 

^Xy(l-3cos2^}  ...  (6). 

For  particles  at  equal  elongations  from  the  sun  and  moon,  we  shall  have  0  the 
same  in  expressions  (a)  and  (6),  and  y  may  be  regarded  as  the  same  without  ma- 
terial error.  For  such  particles,  then,  the  alterations  of  the  gravity,  produced 
by  the  sun  and  moon,  will  bear  the  same  ratio  to  each  other  as  the  quantities 

-5-  and  — i.  Now,  if  we  give  to  m,m'.  a,  a',  their  values,  we  shall  find  that  the 
o*>  a'3 

latter  quantity  is  nearly  three  times  greater  than  the  former.  Accordingly,  the 
effect  of  the  moon's  action,  at  corresponding  elongations  of  the  particles,  and  there- 
fore generally,  is  nearly  three  times  greater  than  that  of  the  sun. 

684.  The  actual  tide  will  be  produced  by  the  joint  action  of  the 
sun  and  moon,  or  it  may  be  regarded  as  the  result  of  the  combina- 
tion of  the  lunar  and  solar  tide  waves. 

At  the  time  of  the  syzigies,  the  action  of  the  sun  and  moon  will 
be  combined  in  producing  the  tides,  both  bodies  tending  to  produce 
high  as  well  as  low  water  at  the  same  places.  But  at  the  quadra- 
tures they  will  be  in  opposition  to  each  other,  the  one  tending  to 
raise  the  surface  of  the  water  where  the  other  tends  to  depress  it, 
and  vice  versa.  The  tides  should,  therefore,  be  much  higher  at 
the  syzigies  than  at  the  quadratures. 

Between  the  syzigies  and  the  quadratures  the  two  bodies  will 
neither  directly  conspire  with  each  other,  nor  directly  oppose  each 
other,  and  tides  of  intermediate  height  will  have  place.  The  points 
of  highest  water  will  also,  in  the  configuration  supposed,  neither 
be  the  vertices  of  the  lunar  nor  of  the  solar  tide  wave,  but  certain 
points  between  them.  This  circumstance  will  occasion  a  variation 
in  the  length  of  the  interval  between  the  time  of  the  moon's  pas- 
sage and  the  time  of  high  water. 

685.  The  effect  of  the  moon's  action  being  to  that  of  the  sun's 
nearly  as  3  to  1,  (683,)  the  spring  tides  will  be  to  the  neap  tides 
nearly  as  2  to  1 .     For,  let  x  =  the  effect  of  the  moon,  and  y  — 
the  effect  of  the  sun :  then  the  ratio  of  x  +  y  to  x  — •  y  will  be  the 
ratio  of  the  heights  of  the  spring  and  neap  tides.     Now, 

*  =  3y,  and  thus  ^=§£±2  =  2. 
x-y      3y-y 

This  result  is  conformable  to  observation. 

686.  The  height  of  the  tide,  as  well  as  the  interval  between  the 
time  of  high  water  and  that  of  the  moon's  meridian  passage,  will 
vary  not  only  with  the  elongation  of  the  moon  from  the  sun,  but 


MODIFICATIONS  OF  THE  GENERAL  PHYSICAL  THEORY.         259 

also  with  the  distance  and  declination  of  the  moon  and  sun.  For, 
expressions  (a)  and  (b)  show  that  the  intensities  of  the  moon's  and 
sun's  actions  vary  inversely  as  the  cube  of  their  distance  ;  and  the 
changes  of  the  declinations  of  the  two  bodies  must  be  attended 
with  a  change  both  in  the  absolute  and  relative  situation  of  the 
vertices  of  the  lunar  and  solar  tide  waves. 

687.  The  laws  of  the  tides,  which  would  obtain  on  the  hypothe- 
sis of  the  earth  being  covered  entirely  with  water,  are  found  to 
correspond  only  partially  with  those  of  the  actual  tides.     The 
continents  have  a  material  influence  upon  the  formation  and  pro- 
pagation of  the  tide  wave. 

688.  Professor  Whewell  infers,  from  a  careful  discussion  of  a 
great  number  of  observations  upon  the  tides,  that  the  tide  of  the 
Atlantic  Ocean  is,  for  the  most  part,  produced  by  a  derivative  tide 
wave,  sent  off  from  the  great  wave  which  in  the  Southern  Ocean 
follows  the  moon  in  its  diurnal  motion  around  the  earth.     This 
wave  advances  more  rapidly  in  the  open  sea  than  along  the  coasts, 
where  it  meets  with  obstructions. 

Where  portions  of  the  tide  wave,  extending  from  one  point  of 
the  coast  to  another,  become  detached,  and  advance  into  a  narrow 
space,  particularly  high  tides  will  occur.    In  this  way  (as  it  is  sup 
posed)  it  happens  that  the  tide  rises  at  certain  places  in  the  Bay 
of  Fundy,  to  the  height  of  60  or  70  feet. 

689.  In  channels  peculiar  tides  occur  in  consequence  of  the 
meeting  of  the  waves  which  enter  the  channels  at  their  two  ex- 
tremities. Where  the  two  waves  meet  in  the  same  state,  unusually 
high  tides  occur.     This  is  observed  to  be  the  case  at  some  points 
in  the  Irish  Channel.    In  the  port  of  Batsha,  in  Tonquin,  the  tides 
arrive  by  two  channels,  of  such  lengths  that  the  two  waves  meet 
in  opposite  states,  or  that  the  flood  tide  arrives  by  one  channel  just 
as  the  ebb  tide  begins  to  leave  by  the  other,  and  the  consequence 
is  that  there  is  neither  high  nor  low  water. 

This  is  the  case  when  the  moon  is  in  the  equator.  When  she 
has  a  northern  or  southern  declination,  there  is  a  small  rise  and 
fall  of  the  water  once  in  a  lunar  day,  owing  to  the  inequality  of  the 
morning  and  evening  tides  of  the  open  sea. 

690.  Lakes  and  inland  seas  have  no  perceptible  tides,  for  the 
reason  that  their  extent  is  not  sufficient  to  admit  of  any  sensible 
inequality  of  gravity,  as  the  result  of  the  action  of  the  moon. 

691.  The  tides  experienced  in  rivers  and  seas  communicating 
with  the  ocean,  are  not  produced  by  the  direct  actions  of  the  sun 
and  moon,  but  are  waves  propagated  from  the  great  wave  of  the 
open  sea. 

In  rivers  of  considerable  length,  the  ascending  tides  are  encoun- 
tered by  those  which  are  returning,  so  that  a  great  variety  of  tides 
occur  along  their  shores. 

692.  The  mean  interval  between  noon  and  the  time  of  high 
water  at  any  port,  on  the  day  of  new  or  full  moon,  is  called  the 


260  OF  THE  TIDES. 

Establishment  of  that  port.  It  will  be,  approximately,  the  inter- 
val between  the  time  of  the  meridian  passage  of  the  moon  and  the 
time  of  high  water  on  any  day  of  the  month.  To  obtain  this  in- 
terval for  a  given  day  more  nearly,  it  is  necessary  to  correct  the 
establishment  for  the  effects  of  the  change  of  the  distance  and  de- 
clination of  the  sun  and  moon,  and  of  the  change  in  the  elongation 
of  the  moon  from  the  sun.  When  it  has  been  determined,  by  add- 
ing it  to  the  time  of  the  meridian  passage  of  the  moon,  we  have  the 
time  of  the  next  high  water. 


PART    IV. 

ASTRONOMICAL    PROBLEMS. 


EXPLANATIONS  OF  THE  TABLES. 

THE  Tables  which  form  a  part  of  this  work,  and  which  are  em 
ployed  in  the  resolution  of  the  following  Problems,  consist  of  Ta- 
bles of  the  Sun,  Tables  of  the  Moon,  Tables  of  the  Mean  Places 
of  some  of  the  Fixed  Stars,  Tables  of  Corrections  for  Refraction, 
Aberration, and  Nutation,  and  Auxiliary  Tables. 

The  Tables  of  the  Sun,  which  are  from  XVII  to  XXXIV,  in- 
elusive,  are,  for  the  most  part,  abridged  from  Delambre's  Solar  Ta- 
bles. The  mean  longitudes  of  the  sun  and  of  his  perigee  for  the 
beginning  of  each  year,  found  in  Table  XVIII,  have  been  com- 
puted from  the  formulae  of  Prof.  Bessel,  given  in  the  Nautical  Al- 
manac of  1837.  The  Table  of  the  Equation  of  Time  was  reduced 
from  the  table  in  the  Connaissance  des  Terns  of  1810,  which  is 
more  accurate  than  Delambre's  Table,  this  being  in  some  instances 
liable  to  an  error  of  2  seconds.  The  Table  of  Nutation  (Table 
XXVII)  was  extracted  from  Francceur's  Practical  Astronomy. 
The  maximum  of  nutation  of  obliquity  is  taken  at  9". 25.  The 
Tables  of  the  Sun  will  give  the  sun's  longitude  within  a  frac- 
tion of  a  second  of  the  result  obtained  immediately  from  De- 
lambre's Tables,  as  corrected  by  Bessel.  The  Tables  of  the 
Moon,  which  are  from  XXXIV  to  LXXXV,  inclusive,  are 
abridged  and  computed  from  Burckhardt's  Tables  of  the  Moon. 
To  facilitate  the  determination  of  the  hourly  motions  in  longi- 
tude and  latitude,  the  equations  of  the  hourly  motions  have  all 
been  rendered  positive,  like  those  of  the  longitude.  Some  few  new 
tables  have  been  computed  for  the  same  purpose.  The  longitude 
and  hourly  motion  in  longitude  will  very  rarely  differ  from  the  re- 
sults of  Burckhardt's  Tables  more  than  0".5,  and  never  as  much 
as  1 ' .  The  error  of  the  latitude  and  hourly  motion  in  latitude  will 
be  still  less.  The  other  tables  have  been  taken  from  some  of  the 
most  approved  modern  Astronomical  Works.  (For  the  principles 
of  the  construction  ofjthe  Tables,  see  Chap.  IX.) 

Before  entering  upon  the  explanation  of  each  of  the  tables,  it 
will  be  proper  to  define  a  few  terms- that  will  be  made  use  of  in  the 
sequel. 

The  given  quantity  with  which  a  quantity  is  taken  from  a  table, 
is  called  the  A  rgument  of  this  quantity. 


£62  ASTR01SOMIC.A.L  PROBLEMS. 

The  angular  arguments  are  expressed  in  some  of  the  tables  ac- 
cording to  the  sexagesimal  division  of  the  circle.  In  others,  they 
are  given  in  parts  of  the  circle  supposed  to  be  divided  into  100, 
1000,  or  10000,  &c.,  parts. 

Tables  are  of  Single  or  Double  Entry,  according  as  they  con- 
tain one  or  two  arguments.  The  Epoch  of  a  table  is  the  instant 
of  time  for  which  the  quantities  given  by  the  table  are  computed. 
By  the  Epoch  of  a  quantity,  is  meant  the  value  of  the  quantity 
found  for  some  chosen  epoch,  from  which  its  value  at  other  epochs 
is  to  be  computed  by  means  of  its  known  rate  of  variation. 

Table  I,  contains  the  latitudes  and  longitudes  from  the  meridian 
of  Greenwich,  of  various  conspicuous  places  in  different  parts  of 
the  earth.  The  longitudes  serve  to  make  known  the  time  at  any 
one  of  the  places  in  the  table,  when  that  at  any  of  the  others  is 
given.  The  latitude  of  a  place  is  an  important  element  in  various 
astronomical  calculations. 

Table  II,  is  a  table  of  the  Elements  of  the  Orbits  of  the  Planets, 
with  their  secular  variations,  which  serve  to  make  known  the  ele- 
ments at  any  given  epoch  different  from  that  of  the  table.  From 
these  the  elliptic  places  of  the  planets  at  the  given  epoch  may  be 
computed. 

Table  III,  is  a  similar  table  for  the  Moon. 

Tables  IV,  V,  VI,  VII,  require  no  explanation. 

Table  VIII,  gives  the  mean  Astronomical  Refractions  ;  that  is, 
the  refractions  which  have  place  when  the  barometer  stands  at  30 
inches,  and  the  thermometer  of  Fahrenheit  at  50°. 

Table  IX,  contains  the  corrections  of  the  Mean  Refractions  for 
-f-1  inch  in  the  barometer,  and—  1°  in  the  thermometer,  from 
which  the  corrections  to  be  applied,  at  any  observed  height  of  the 
barometer  and  thermometer,  are  easily  derived. 

Table  X,  gives  the  Parallax  of  the  Sun  for  any  given  altitude  on 
a  given  day  of  the  year  ;  for  reducing  a  solar  observation  made  at 
the  surface  of  the  earth  to  what  it  would  have  been,  if  made  at  the 
centre. 

Table  XI,  is  designed  to  make  known  the  Sun's  Semi-diurnal 
Arc,  answering  to  any  given  latitude  and  to  any  given  declination 
of  the  sun ;  and  thus  the  time  of  the  sun's  rising  and  setting,  and 
the  length  of  the  day. 

Table  XII,  serves  to  make  known  the  value  of  the  Equation  of 
Time,  with  its  essential  sign,  which  is  to  be  applied  to  the  apparent 
time  to  convert  it  into  the  mean.  If  the  sign  of  the  equation  taken 
from  the  table  be  changed,  it  will  serve  for  the  conversion  of  mean 
time  into  apparent.  This  table  is  constructed  for  the  year  1840. 

Table  XIII,  is  to  be  used  in  connection  with  Table  XII,  when 
the  given  date  is  in  any  other  year  than  1840.  It  furnishes  the 
Secular  Variation  of  the  Equation  of  Time,  from  which  the  pro- 
portional part  of  its  variation  in  the  interval  between  the  given  date 
and  the  epoch  of  Table  XII  is  easily  derived. 


EXPLANATION   OF  THE  TABLES.  263 

Table  XIV,  contains  certain  other  Corrections  to  be  applied  to 
the  equation  of  time  taken  from  Table  XII,  when  its  exact  value, 
to  within  a  small  fraction  of  a  second,  is  desired. 

Table  XV,  gives  the  Fraction  of  the  Year  corresponding  to  each 
date.  This  table  is  useful  when  quantities  vary  by  known  and  uni- 
form degrees,  in  deducing  their  values  at  any  assumed  time  from 
their  values  at  any  other  time. 

Table  XVI,  is  for  converting  Hours,  Minutes,  and  Seconds  into 
decimal  parts  of  a  Day. 

Table  XVII,  is  for  converting  Minutes  and  Seconds  of  a  degree 
into  the  decimal  division  of  the  same.  It  will  also  serve  for  the 
conversion  of  minutes  and  seconds  of  time  into  decimal  parts  of  an 
hour. 

The  last  two  tables  will  be  found  frequently  useful  in  arithmeti- 
cal operations 

Table  XVIII,  is  a  table  of  Epochs  of  the  Sun's  Mean  Longi- 
tude, of  the  Longitude  of  the  Perigee,  and  of  the  Arguments  for 
finding  the  small  equations  of  the  Sun's  place.  They  are  all  cal- 
culated for  the  first  of  January  of  each  year,  at  mean  noon  on  the 
meridian  of  Greenwich.  Argument  I.  is  the  mean  longitude  of  the 
Moon  minus  that  of  the  Sun ;  Argument  II.  is  the  heliocentric 
longitude  of  the  Earth ;  Argument  III.  is  the  heliocentric  longi- 
tude of  Venus  ;  Argument  IV.  is  the  heliocentric  longitude  of 
Mars ;  Argument  V.  is  the  heliocentric  longitude  of  Jupiter ;  Ar- 
gument VI.  is  the  mean  anomaly  of  the  Moon ;  Argument  VII.  is 
the  heliocentric  longitude  of  Saturn ;  and  Argument  N  is  the  sup- 
plement of  the  longitude  of  the  Moon's  Ascending  Node.  Argu- 
ment I.  is  for  the  first  part  of  the  equation  depending  on  the  action 
of  the  Moon.  Arguments  I.  and  VI.  are  the  arguments  for  the  re- 
maining part  of  the  lunar  equation.  Arguments  II.  and  III.  are  for 
the  equation  depending  on  the  action  of  Venus ;  Arguments  II. 
and  IV.  for  the  equation  depending  on  the  action  of  Mars ;  Argu- 
ments II.  and  V.  for  the  equation  depending  on  the  action  of  Ju- 
piter ;  and  Arguments  II.  and  VII.  for  the  equation  depending  on 
the  action  of  Saturn.  Argument  N  is  the  argument  for  the  Nuta- 
tion in  longitude  :  it  is  also  the  argument  for  the  Nutation  in  right 
ascension,  and  of  the  obliquity  of  the  ecliptic. 

Table  XIX,  shows  the  Motions  of  the  Sun  and  Perigee,  and  the 
variations  of  the  arguments,  in  the  interval  between  the  beginning 
of  the  year  and  the  first  of  each  month. 

Table  XX,  shows  the  Motions  of  the  Sun  and  Perigee,  and  the 
variations  of  the  arguments  from  the  beginning  of  any  month  to  the 
Beginning  of  any  day  of  the  month  ;  also  the  same  for  Hours. 

Table  XXI,  gives  the  Sun's  Motions  for  Minutes  and  Seconds. 
Tables  XVIII  to  XXI,  inclusive,  make  known  the  mean  longitude 
of  the  Sun  from  the  mean  equinox,  at  any  moment  of  time. 

Table  XXII,  Mean  Obliquity  of  the  Ecliptic  for  the  beginning 


264  ASTRONOMICAL  PROBLEMS. 

of  each  year  contained  in  the  table.  It  is  found  for  any  interme- 
diate time  by  simple  proportion. 

Tables  XXIII,  and  XXIV,  furnish  the  Sun's  Hourly  Motion 
and  Semi-diameter. 

Table  XXV,  is  designed  to  make  known  the  Equation  of  the 
Sun's  Centre.  When  the  equation  has  the  negative  sign,  its  sup- 
plement to  12s.  is  given :  this  is  to  be  added  along  with  the  other 
equations  of  longitude,  and  12s.  are  to  be  subtracted  from  the  sum. 

The  numbers  in  the  table  are  the  values  of  the  equation  of  the 
centre,  or  of  its  supplement,  diminished  by  46". 1.  This  constant 
is  subtracted  from  each  value,  to  balance  the  different  quantities 
added  to  the  other  equations  of  the  longitude,  in  order  to  render 
them  affirmative.  The  epoch  of  this  table  is  the  year  1840. 

Table  XXVI,  gives  the  Secular  Variation  of  the  Equation  of  the 
Sun's  Centre,  from  which  the  proportional  part  of  the  variation  in 
the  interval  between  the  given  date  and  the  year  1840,  may  be 
derived. 

Table  XXVII,  is  for  the  Nutation  in  Longitude,  Nutation  in 
Right  Ascension,  and  Nutation  of  the  Obliquity  of  the  Ecliptic. 
The  nutation  in  longitude  and  nutation  in  right  ascension,  serve  to 
transfer  the  origin  of  the  longitude  and  right  ascension  from  the 
mean  to  the  true  equinox.  And  the  nutation  of  obliquity  serves  to 
change  the  mean  into  the  true  obliquity. 

Tables  XXVIII  to  XXXIII,  inclusive,  give  the  Equations  of 
the  Sun's  Longitude,  due  respectively  to  the  attractions  of  the 
Moon,  Venus,  Jupiter,  Mars,  and  Saturn. 

Table  XXXIV,  is  for  the  variable  part  of  the  Sun's  Aberration. 
The  numbers  have  all  been  rendered  positive  by  the  addition  of 
the  constant  0".3. 

Table  XXXV,  contains  the  Epochs  of  the  Moon's  Mean  Longi- 
tude, and  of  the  Arguments  of  the  equations  used  in  determining 
the  True  Longitude  and  Latitude  of  the  Moon.  They  are  all  cal- 
culated for  the  first  of  January  of  each  year,  at  mean  noon  on  the 
meridian  of  Greenwich.  The  Argument  for  the  Evection  is  di- 
minished by  30' ;  the  Anomaly  by  2°  ;  the  Argument  for  the  Va- 
riation by  9°,  and  the  mean  longitude  by  9°  45'' ;  and  the  Supple- 
ment of  the  Node  is  increased  by  7'.  This  is  done  to  balance  the 
quantities  which  are  added  to  the  different  equations  in  order  to 
render  them  affirmative. 

Tables  XXXVI  to  XL,  inclusive,  give  the  Motions  of  the  Moon, 
and  the  variations  of  the  arguments,  for  Months,  Days,  Hours, 
Minutes,  and  Seconds  ;  and,  together  with  Table  XXXV,  are  for 
finding  the  Moon's  Mean  Longitude  and  the  Arguments,  at  any 
assumed  moment  of  time. 

Tables  XLI  to  LIII,  inclusive,  give  the  various  Equations  of 
the  Moon's  Longitude.  It  is  to  be  observed  with  respect  to  Table 
XLI,  that  the  right  hand  figure  of  the  argument  is  supposed  to  be 
dropped.  But  when  the  greatest  attainable  accuracy  is  desired,  it 


EXPLANATION  OF  THE  TABLES.  265 

can  be  retained,  and  a  cipher  conceived  to  be  written  after  the 
numbers  in  the  columns  of  Arguments  in  the  table.  In  Tables 
L,  LI,  LII,  and  LV,  the  degrees  will  be  found  by  referring  to  the 
head  or  foot  of  the  column.  (See  Problem  II.,  note  2.) 

Table  LIV  is  for  the  Nutation  of  the  Moon's  Longitude. 

Tables  LV  to  LIX,  inclusive,  are  for  finding  the  Latitude  of 
the  Moon. 

Tables  LX  to  LXIII,  inclusive,  are  for  the  Equatorial  Paral- 
lax of  the  Moon. 

Table  LXIV  furnishes  the  Reductions  of  Parallax  and  of  the 
Latitude  of  a  Place.  The  reduction  of  parallax  is  for  obtaining 
the  parallax  at  any  given  place  from  the  equatorial  parallax.  The 
reduction  of  latitude  is  foi  reducing  the  true  latitude  of  a  place,  as 
determined  by  observation,  to  the  corresponding  latitude  on  the 
supposition  of  the  earth  being  a  sphere.  The  ellipticity  to  which 
the  numbers  in  the  table  correspond  is  -g^-g. 

Tables  LXV  and  LXVI,  Moon's  Semi-diameter,  and  the  Aug- 
mentation of  the  Semi-diameter  depending  on  the  altitude. 

Tables  LXVII  to  LXXXV,  inclusive,  are  for  finding  the 
Hourly  Motions  of  the  Moon  in  Longitude  and  Latitude. 

Table  LXXXVI,  Mean  New  Moons,  and  the  Arguments  for  the 
Equations  for  New  and  Full  Moon,  in  January.  The  time  of 
mean  new  moon  in  January  of  each  year  has  been  diminished  by 
15  hours,  the  sum  of  the  quantities  which  have  been  added  to  the 
equations  in  Table  LXXXIX.  Thus,  4h.  20m.  has  been  added 
to  equation  I. ;  lOh.  10m.  to  equation  II.  ;  10m.  to  equation  III.; 
and  20m.  to  equation  IV. 

Tables  LXXXVII  and  LXXXVIII,  are  used  with  the  preced- 
ing in  finding  the  Approximate  Time 'of  Mean  New  or  Full  Moon 
in  any  given  month  of  the  year. 

Table  LXXXIX  furnishes  the  Equations  for  finding  the  Ap- 
proximate Time  of  New  or  Full  Moon. 

Table  XC  contains  the  Mean  Right  Ascensions  and  Declina- 
tions of  50  principal  Fixed  Stars,  for  the  beginning  of  the  year 
1840,  with  their  Annual  Variations. 

Table  XCI  is  for  finding  the  Aberration  and  Nutation  of  the 
Stars  in  the  preceding  catalogue. 

Table  XCII  contains  the  Mean  Longitudes  and  Latitudes  of 
some  of  the  principal  Fixed  Stars,  for  the  beginning  of  the  year 
1840,  with  their  Annual  Variations. 

Tables  XCIII,  XCIV,  XCV,  Second,  Third,  and  Fourth 
Differences.  These  tables  are  given  to  facilitate  the  determina- 
tion, from  the  Nautical  Almanac,  of  the  moon's  longitude  or  lati- 
tude for  any  time  between  noon  and  midnight. 

Table  XCVI,  Logistical  Logarithms.  This  table  is  convenient 
in  working  proportions,  when  the  terms  are  minutes  and  seconds, 
or  degrees  and  minutes,  or  hours  and  minutes, — especially  when 
the  first  term  is  Ih.  or  60m. 

34 


266  ASTRONOMICAL  PROBLEMS. 

To  find  the  logistical  logarithm  of  a  number  composed  of  min- 
utes and  seconds,  or  degrees  and  minutes,  of  an  arc ;  or  of  min- 
utes and  seconds,  or  hours  and  minutes,  of  time. 

1.  If  the  number  consists  of  minutes  and  seconds,  at  the  top  of 
the  table  seek  for  the  minutes,  and  in  the  same  column  opposite 
the  seconds  in  the  left-hand  column  will  be  found  the  logistical 
logarithm. 

2.  If  the  number  is  composed  of  hours  and  minutes,  the  hours 
must  be  used  as  if  they  were  minutes,  and  the  minutes  as  if  they 
were  seconds. 

3.  If  the  number  is  composed  of  degrees  and  minutes,  the  de- 
grees must  be  used  as  if  they  were  minutes,  and  the  minutes  as  if 
they  were  seconds. 

To  find  the  logistical  logarithm  of  a  number  less  than  3600. 

Seek  in  the  second  line  of  the  table  from  the  top  the  number 
next  less  than  the  given  number,  and  the  remainder,  or  the  com- 
plement to  the  given  number,  in  the  first  column  on  the  left :  then 
in  the  column  of  the  first  number,  and  opposite  the  complement, 
will  be  found  the  logistical  logarithm  of  the  sum.  Thus,  to  ob- 
tain the  logarithm  of  1531,  we  seek  for  the  column  of  1500,  and 
opposite  31  we  find  3713. 


PROBLEM  I. 

\ 

To  work,  by  logistical  logarithms,  a  proportion  the  terms  of  which 
are  degrees  and  minutes,  or  minutes  and  seconds,  of  an  arc ;  or 
hours  and  minutes,  or  minutes  and  seconds,  of  time. 

With  the  degrees  or  minutes  at  the  top,  and  minutes  or  seconds 
at  the  side,  or  if  a  term  consists  of  hours  and  minutes,  or  minutes 
and  seconds,  with  the  hours  or  minutes  at  the  top,  and  minutes 
or  seconds  at  the  side,  take  from  Table  XCVI.  the  logistical  loga- 
rithms of  the  three  given  terms  ;  add  together  the  logistical  loga- 
rithms of  the  second  and  third  terms  and  the  arithmetical  comple- 
ment of  that  of  the  first  term,  rejecting  10  from  the  index.*  The 
result  will  be  the  logistical  logarithm  of  the  fourth  term,  with 
which  take  it  from  the  table. 

Note  1.  The  logistical  logarithm  of  60'  is  0. 

Note  2.  If  the  second  or  third  term  contains  tenths  of  seconds, 
(or  tenths  of  minutes,  when  it  consists  of  degrees  and  minutes,) 
and  is  less  than  6',  or  6°,  multiply  it  by  10,  and  employ  the  loga- 
rithm of  the  product  in  place  of  that  of  the  term  itself.  The 

*  Instead  of  adding  the  arithmetical  complement  of  the  ogarithm  of  the  first 
term,  the  logarithm  itself  may  be  subtracted  from  the  sum  of  the  logarithms  of  the 
other  two  terms. 


TO  TAKE  OUT  A  QUANTITY  FROM  A  TABLE.        26*7 

result  obtained  by  the  table,  divided  by  10,  will  be  the  fourth  term 
of  the  proportion,  and  will  be  exact  to  tenths. 

Note  3.  If  none  of  the  terms  contain  tenths  of  minutes  or  sec- 
onds, and  it  is  desired  to  obtain  a  result  exact  to  tenths,  diminish 
the  index  of  the  logistical  logarithm  of  the  fourth  term  by  1,  and 
cut  off  the  right-hand  figure  of  the  number  found  from  the  table, 
for  tenths. 

Exam.  1.  When  the  moon's  hourly  motion  is  30'  12",  what  is 
its  motion  in  16m.  24s.  ? 

As  60m.       ,''..-  •-/••"     .      ;'^        0 

:    30'  12"   .        '.'     r/v  ;      .  2981 

:  :    16m.  24s.  .  5633 


:    8' 15"      .         .      '  V4      .  8614 

2.  If  the  moon's  declination  change  1°  31'  in  12  hours,  what 
will  be  the  change  in  7h.  42m.  ? 

f  As  12h.    .     ''*         .  ar.  co.  9.3010 

:    1°  31'        **?'.'     .      &     1.5973 

::    7h.  42m.    y  .  ^  'Y  ./' ./    8917 

:    0°  58;      ]  'I   [     .         .     1.7900 

3.  When  the  moon's  hourly  motion  in  latitude  is  2'  26".8,  what 
is  its  motion  in  36m.  22s.  ? 

2'  26".8 
60 


As  60m.  .    *,,-••  0 
1468   .   V  :  1468"  .    .  3896 
:  :  36m.  22s.   .  2174 

:  890"  :'*  '•      .  6070 

Ans.  1'  29". 0. 

4.  When  the  sun's  hourly  motion  in  longitude  is  2'  28",  what 
is  its  motion  in  49m.  11s.  ?  Ans.  2;  1". 

5.  If  the  sun's  decimation  change  16'  33"  in  24  hours,  what 
will  be  the  change  in  14h.  18m.  ?  Ans.  9'  52". 

6.  If  the  moon's  declination  change  54".7  in  one  hour,  what  will 
be  the  change  in  52m.  18s.  ?  Ans.  47".7. 


PROBLEM  II. 

To  take  from  a  table  the  quantity  corresponding  to  a  given  value 
of  the  argument,  or  to  given  values  of  the  arguments  of  the 
table. 


268  ASTRONOMICAL  PROBLEMS. 

Case  1.  When  quantities  are  given  in  the  table  for  each  sign 
and  degree  of  the  argument. 

With  the  signs  of  the  given  argument  at  the  top  or  bottom,  and 
the  degrees  at  the  side,  (at  the  left  side,  if  the  signs  are  found  at 
the  top ;  at  the  right  side,  if  they  are  found  at  the  bottom,)  take  out 
the  corresponding  quantity.  Also  take  the  difference  between  this 
quantity  and  the  next  following  one  in  the  table,  and  say,  60' :  this 
difference  :  :  odd  minutes  and  seconds  of  given  argument :  a  fourth 
term.  This  fourth  term,  added  to  the  quantity  taken  out,  when  the 
quantities  in  the  table  are  increasing,  but  subtracted  when  they  are 
decreasing,  will  give  the  required  quantity. 

Note  1 .  When  the  quantities  change  but  little  from  degree  to 
degree  of  the  argument,  the  required  quantity  may  often  be  esti- 
mated, without  the  trouble  of  stating  a  proportion. 

Note  2.  In  some  of  the  tables  the  degrees  or  signs  of  the  quan- 
tity sought,  are  to  be  had  by  referring  to  the  head  or  foot  of  the  col- 
umn in  which  the  minutes  and  seconds  are  found.  (See  Tables 
L,  LI,  LII,  and  LV.)  The  degrees  there  found  are  to  be  taken, 
if  no  horizontal  mark  intervenes ;  otherwise,  they  are  to  be  in- 
creased or  diminished  by  1°,  or  2°,  according  as  one  or  two  marks 
intervene.  They  are  to  be  increased,  or  diminished,  according  as 
their  number  is  less  or  greater  than  the  number  of  degrees  at  the 
other  end  of  the  column. 

Note  3:  If,  as  is  the  case  with  some  of  the  tables,  the  quantities 
in  the  table  have  an  algebraic  sign  prefixed  to  them,  neglect  the 
consideration  of  the  sign  in  determining  the  correction  to  be  applied 
to  the  quantity  first  taken  out,  and  proceed  according  to  the  rule 
above  given.  The  result  will  have  the  sign  of  the  quantity  first 
taken  out.  It  is  to  be  observed,  however,  that  if  the  two  consecu- 
tive quantities  chance  to  have  opposite  signs,  their  numerical  sum 
is  to  be  taken  instead  of  their  difference ;  also  that  the  quantity 
sought  will,  in  every  such  instance,  be  the  numerical  difference 
between  the  correction  and  the  quantity  first  taken  out,  and,  ac- 
cording as  the  correction  is  less  or  greater  than  this  quantity,  is  to 
be  affected  with  the  same  or  the  opposite  sign. 

Exam.  1.  Given  the  argument  7s-  6°  24'  36",  to  find  the  corre- 
sponding quantity  in  Table  L. 

7s-  6°  gives  0°  43'  17" .4. 

The  difference  between  0°  43'  17"  .4  and  the  next  following  quan- 
tity in  the  table  is  1'  7".3. 

60'  :  1'  7".3  :  :  24'  36"  :  27".6.* 

*  The  student  can  work  the  proportion,  either  by  the  common  method,  or  by  lo- 
gistical logarithms,  as  he  may  prefer.  In  working  this  and  all  similar  proportions 
by  the  arithmetical  method,  the  seconds  of  the  argument  may  be  converted  into 
the  equivalent  decimal  part  of  a  minute  by  means  of  Table  XVII,  (using  the  sec- 
onds  as  if  they  were  minutes.)  It  will  be  sufficient  to  take  the  fraction  to  the 
nearest  tenth. 


TO  TAKE  OUT  A  QUANTITY  FROM  A  TABLE.         269 

From  0°  43'  17"  A 
Take  27  .6 


0  42   49  .8 

2.  Given  the  argument  2s-  18°  41'  20",  to  find  the  corresponding 
quantity  in  Table  XXV. 

2s-  18°  gives  1°  52'  32".5. 

The  difference  between  1°  52'  32".  5  and  the  next  following 
quantity  in  the  table  is  21".8. 

60'  :  21".8  :  :  41'  20"  :  15".0. 

To     1°  52'  32".5 
Add  15  .0 


1    52  47  .5 

3.  Given  the  argument  9*  2°  13'  33",  to  find  the  correspond- 
ing quantity  in  Table  XII. 

9s-  2°  gives  29.8s. 

The  arithmetical  sum  of  29.8s.  and  the  next  following  quantity 
in  the  table  is  30.4s. 

60'  :  30.4s.  :  :  13°  33'  :  6.9s. 

From     29.8s. 
Take       6.9 

22.9s. 
Ans.  —  22.9s. 

4.  Given  the  argument  5s-  8°  14'  52",  to  find  the  corresponding 
quantity  in  Table  LII.  Ans.  12'  36".0. 

5.  Given  the  argument  118>  11°  23'  10",  to  find  the  correspond- 
ing quantity  in  Table  LVI.  Ans.  1 17  48'  .0. 

6.  Given  the  argument  0s-  26°  20',  to  find  the  corresponding 
quantity  in  Table  XII.  Ans.  —  4 P.O. 

Case  2.  When  the  argument  changes  in  the  table  by  more  or 
less  than  1°;  or  when  it  is  given  in  lower  denominations  than 
signs. 

Take  out  of  the  table  the  quantity  answering  to  the  number  in 
the  column  of  arguments  next  less  than  the  given  argument.  Take 
the  difference  between  this  quantity  and  the  next  following  one, 
and  also  the  difference  of  the  consecutive  values  of  the  argument 
inserted  in  the  table,  and  say,  difference  of  arguments  :  difference 
of  quantities  :  :  excess  of  the  given  argument  over  the  value  next 
less  in  the  table  :  a  fourth  term.  This  fourth  term  applied  to  the 
quantity  first  taken  out,  according  to  the  rule  given  in  the  prece- 
ding case,  will  give  the  quantity  sought. 

Note.  In  some  of  the  tables  the  columns  entitled  Diff.  are  made 
up  of  the  differences  answering  to  a  difference  of  10'  in  the  argu- 
ment. In  obtaining  quantities  from  these  tables,  it  will  be  found 
more  convenient  to  take  for  the  first  and  second  terms  of  the  pro- 


270  ASTRONOMICAL  PROBLEMS. 

portion,  respectively,  10',  and  the  difference  furnished  by  the  table, 
and  work  the  proportion  by  the  arithmetical  method.  (See  note  at 
bottom  of  page  268.) 

Exam.  1.  Given  the  argument  0s-  24°  42'  15",  to  find  the  cor- 
responding quantity  in  Table  LI. 

0s-  24°  30'  gives  9°  47'  14".3. 

The  difference  between  9°  47'  14".3  and  the  next  following 
quantity  =  3  x  63".0  =  189".0.  The  argument  changes  by  30'. 
And  the  excess  of  0s-  24°  42'  15"  over  0s-  24°  30',  is  12  15".  Thus, 

30'  :  189".0  :  :  12'  15"  :  77".2. 

But  the  correction  may  be  found  more  readily  by  the  following 
proportion : 

10'  :  63".0  :  :  12'.25  :  77".2. 

To     9°  47'  14"  .3 
Add  77  .2 


*  9   48  31  .5 

2.  Given  the  argument  1°  12',  to  find  the  corresponding  quan- 
tity in  Table  VIII. 

1°  10'  gives  23'  13", 
and  5'  :  33"  :  :  2'  :  13"  the  correction. 

From         23'  13" 
Take  13 

23    0 

3.  Given  the  argument  6s-  6°  7'  23",  to  find  the  corresponding 
quantity  in  Table  LV.  Ans.  90°  20'  53".5. 

4.  Given  the  argument  49°  27',  to  find  the  corresponding  quan- 
tity in  Table  LXIV.  Ans.  11'  19".8. 

Case  3.   When  the  argument  is  given  in  the  table  in  hundredth, 
thousandth,  or  ten  thousandth  parts  of  a  circle. 

The  required  quantity  can  be  found  in  this  case  by  the  same 
rule  as  in  the  preceding ;  but  it  can  be  had  more  expeditiously  by 
observing  the  following  rules.  If  the  argument  varies  by  10,  mul- 
tiply the  difference  of  the  quantities  between  which  the  required 
quantity  lies  by  the  excess  of  the  given  argument  over  the  next  less 
value  in  the  table,  and  remove  the  decimal  point  one  figure  to  the 
left ;  the  result  will  be  the  correction  to  be  applied  to  the  quantity 
taken  out  of  the  table.  The  same  rule  will  apply  in  taking  quan- 
tities from  tables  in  which  the  differences  answering  to  a  change  of 
10  in  the  argument  are  given,  although  the  argument  should  actu- 
ally change  by  50  or  100.  If  the  argument  changes  by  100,  mul 
tiply  as  above,  and  remove  the  decimal  point  two  figures  to  the  left. 
When  the  common  difference  of  the  arguments  is  5,  proceed  as  if 
it  were  10,  and  double  the  result.  In  like  manner,  when  the  com- 
mon difference  is  50,  proceed  as  if  it  were  100,  and  double  the 
result. 


TO  TAKE  OUT  A  QUANTITY  FROM  A  TABLE.         271 

Exam.  1.  Given  the  argument  973,  to  find  the  corresponding 
quantity  in  Table  XLV  column  headed  13. 

970  gives  23".5. 
The  difference  is  1". 2,  and  the  excess  3. 

1".2  From    23".5 

3  Take          .4 


Corr.     .36  23  .1 

2.  Given  the  argument  4834,  to  find  the  corresponding  quantity 
m  Table  XLII,  column  headed  5. 

4800  gives  2'  3".7. 

The  difference  is  6".8,  and  the  excess  34. 
6".8 
34 

From   2'  3". 7 

2.312  Take         2  .3 


2    1  .4 

3.  Given  the  argument  5444,  to  find  the  corresponding  quan- 
tity in  Table  XLI.  Ans.  15'  37".7. 

4.  Given  the  argument  4225,  to  find  the  corresponding  quan- 
tity in  Table  XLIII,  column  headed  8.  Ans.  0'  47". 2. 

Case  4.  When  the  table  is  one  of  double  entry,  or  quantities  are 
taken  from  it  by  means  of  two  arguments. 

Take  out  of  the  table  the  quantity  answering  to  the  values  of 
the  arguments  of  the  table  next  less  than  the  given  values  ;  and 
find  the  respective  corrections  to  be  applied  to  it,  due  to  the  ex- 
cess of  the  given  value  of  each  argument  over  the  next  less  value 
in  the  table,  by  the  general  rule  in  the  preceding  case.  These 
corrections  are  to  be  added  to  the  quantity  taken  out,  or  subtracted 
from  it,  according  as  the  quantities  increase  or  decrease  with  the 
arguments. 

Note  1.  If  the  tenths  of  seconds  be  omitted,  the  corrections 
above  mentioned  can  be  estimated  without  the  trouble  of  stating  a 
proportion,  or  performing  multiplications. 

Note  2.  The  rule  above  given  may,  in  some  rare  instances,  give 
a  result  differing  a  few  tenths  of  a  second  from  the  truth.  The 
following  rule  will  furnish  more  exact  results.  Find  the  quanti- 
ties corresponding,  respectively,  to  the  value  of  the  argument  at 
the  top  next  less  than  its  given  value  and  the  other  given  argu- 
ment, and  to  the  value  next  greater  and  the  other  given  argument. 
Take  the  difference  of  the  quantities  found,  and  also  the  difference 
of  the  corresponding  arguments  at  top,  and  say,  difference  of  argu- 
ments :  difference  of  quantities  :  :  excess  of  given  value  of  the 
argument  at  the  top  over  its  next  less  value  in  the  table  :  a  fourth 
term.  This  fourth  term  added  to  the  quantity  first  found,  if  it  is 
less  than  the  other,  but  subtracted  from  it,  if  it  is  greater,  will  give 
the  required  quantity.  The  error  of  the  first  rule  may  be  dimin- 


272  ASTRONOMICAL  PROBLEMS. 

ished  without  any  extra  calculation,  by  attending  to  the  difference 
of  the  quantities  answering  to  the  value  of  the  argument  at  the 
side  next  greater  than  its  given  value  and  the  values  of  the  other 
argument  between  which  its  given  value  lies. 

Examv  1 .  Given  the  argument  64  at  the  top  and  77  at  the  side, 
to  find  the  corresponding  quantity  in  Table  LXXXI. 

50  and  70  give  47".7. 

The  difference  between  47".7  and  the  next  quantity  below  it 
is  I" A.  The  excess  of  77  over  70  is  7,  and  the  argument  at  the 
side  changes  by  10. 

I"  A 
7 

From    47".7 
Corr.  due  excess  7,    .98,  or  1".0.          Take      1.0 

Quantity  corresponding  to  50  and  77,      46  .7 
The  difference  between  47". 7  and  the  adjacent  quantity  in  the 
next  column  on  the  right  is  3". 3.     The  excess  of  64  over  50  is  14, 
and  the  argument  at  the  top  changes  by  50. 
3".3 
14 


.462 
2 

From    46".7 


Corr.  due  excess  14,   .924  Take      0.9 


45  .8 

2.  Given  the  argument  223  at  the  top  and  448  at  the  side,  to 
find  the  corresponding  quantity  in  Table  XXX. 

220  and  440  give  16".0. 

The  difference  between  16".0  and  the  quantity  next  below  it 
is  2".2. 

2".2 
8 

2)  1.76 

From    16/;.0 
Corr.  for  excess  8,         .88,  or  0".9.     Take      0  .9 

Quantity  corresponding  to  220  and  448,  15  .1 
The  difference  between  16".0  and  the  adjacent  quantity  in  the 
next  column  on  the  right  is  0".7. 
0;/.7 
3 

To     15".l 

Corr.  for  excess  3,       £1  Add        .2 

15.3 


TO  CONVERT  DEGREES,  MINUTES,  ETC.,  INTO  TIME.  273 

3.  Given  the  argument  472  at  the  top  and  786  at  the  side,  to 
find  the  corresponding  quantity  in  Table  XXXI. 

AJIS.  9".7. 

4.  Given  the  argument  620  at  the  top  and  367  at  the  side,  to 
find  the  corresponding  quantity  in  Table  LXXXI. 

Ans.  55".2. 

5.  Given  the  argument  348  at  the  top  and  932  at  the  side,  to 
find  (by  the  rule  given  in  Note  2)  the  corresponding  quantity  in 
Table  XXXII.  Ans.  15".4. 


PROBLEM  III. 

To  convert  Degrees,  Minutes,  and  Seconds  of  the  Equator  into 
Hours,  Minutes,  fyc.,  of  Time. 

Multiply  the  quantity  by  4,  and  call  the  product  of  the  seconds, 
thirds  ;  of  the  minutes,  seconds  j  and  of  the  degrees,  minutes. 
Exam.  1.  Convert  83°  II7  52"  into  time. 
83°  11'  52" 
4 


5h-  32m-  47B-  28'" 
2.  Convert  34°  57'  46"  into  time. 

Ans.  2h.  I9m.  51sec.  4"'. 


PROBLEM  IV. 

To  convert  Hours,  Minutes,  and  Seconds  of  Time  into  Degrees, 
Minutes,  and  Seconds  of  the  Equator. 

Reduce  the  hours  and  minutes  to  minutes  :  divide  by  4,  and 
call  the  quotient  of  the  minutes,  degrees  ;  of  the  seconds,, minutes  ; 
and  multiply  the  remainder  by  15,  for  the  seconds. 

Exam.  1.  Convert  7h.  9m.  34sec.  into  degrees,  &c. 
7h.  9m.  34«. 

60 


4  )  429  34 


107°  23'  30" 
2.  Convert  1  Ih.  24m.  45s.  into  degrees,  &c. 

Ans.  171°  11' 
35 


274  ASTRONOMICAL  PROBLEMS. 


PROBLEM   V. 

The  Longitudes  of  two  Places,  and  the  Time  at  one  of  them 
being  given,  to  find  the  corresponding  Time  at  the  other. 

When  the  given  time  is  in  the  morning,  change  it  to  astronomi- 
cal time,  by  adding  12  hours,  and  diminishing  the  number  of  the 
day  by  a  unit.  When  the  given  time  is  in  the  evening,  it  is  al- 
ready in  astronomical  time. 

Find  the  difference  of  longitude  of  the  two  places,  by  taking  the 
numerical  difference  of  their  longitudes,  when  these  are  of  the 
same  name,  that  is,  both  east  or  both  west ;  and  the  sum,  when 
they  are  of  different  names,  that  is,  one  west  and  the  other  east. 
When  one  of  the  places  is  Greenwich,  the  longitude  of  the  other 
is  the  difference  of  longitude. 

Then,  if  the  place  at  which  the  time  is  required  is  to  the  east 
of  the  place  at  which  the  time  is  given,  add  the  difference  of  longi- 
tude, in  time,  to  the  given  time  ;  but,  if  it  is  to  the  west,  subtract 
the  difference  of  longitude  from  the  given  time.  The  sum  or  re- 
mainder will  be  the  required  time. 

Note.  The  longitudes  used  in  the  following  examples,  are  given 
in  Table  I. 

Exam.  1.    When  it  is  October  25th,  3h.  13m.  22sec.  A.  M.  at 
Greenwich,  what  is  the  time  as  reckoned  at  New  York? 
Time  at  Greenwich,  October,  24d>  15h-  13m-  228- 
Diff.  of  Long.         ...  4     56       4 

Time  at  New  York        .         .  24    10     17     18P.M. 

2.  When  it  is  June  9th,  5h.  25m.  lOsec.  P.  M.  at  Washington, 
what  is  the  corresponding  time  at  Greenwich  ? 

Time  at  Washington,  June,         9d-  5h-  25m-  108- 
Diff.  of  Long.         ...  586 

Time  at  Greenwich        .         .     9  10    33      16P.M. 

3.  When  it  is  January  15th,  2h.  44m.  23sec.  P.  M.  at  Paris, 
what  is  the  time  at  Philadelphia  ? 

Longitude  of  Paris         .      ^f  1     Oh-  9m-  21fl.6    E. 
Do.        of  Philadelphia,     .         5    0     39  .6   W. 

5  10       1.2 

Time  at  Paris,  January,         .       15d  2h-  44m-  23'- 
Diff.  of  Long.        .         .     •£»".]         5     10       1 

Time  at  Philadelphia,         "V   14  21     34     22 
Or  January  15th,  9h.  34m.  22sec.  A.  M. 

4.  When  it  is  Marches  1st,  8h.  4m.  21  sec.  P.  M.  at  New  Haven, 
•what  is  the  corresponding  time  at  Berlin  ? 

Ans.  April  1st,  Ih.  49m.  43sec.  A.  M. 


TO  CONVERT  APPARENT  INTO  MEAN  TIME.  275 

5.  When  it  is  August  10th,  lOh.  32m.  Msec.  A.  M.  at  Boston, 
what  is  the  time  at  New  Orleans  ? 

Ans.  Aug.  10th,  9h.  16m.  4sec.  A.  M. 

6.  When  it  is  noon  of  the  23d  of  December  at  Greenwich,  what 
is  the  time  at  New  York  ? 

Ans,   Dec.  23d,  7h.  3m.  55sec.  A.  M 


PROBLEM  VI. 

The  Apparent  Time  being  given,  tojind  the  corresponding  Mean 
Time  ;  or  the  Mean  Time  being  given  tojind  the  Apparent. 

When  the  given  time  is  not  for  the  meridian  of  Greenwich,  re- 
duce it  to  that  meridian  by  the  last  problem.  Then  find  by  the 
tables  the  sun's  mean  longitude  corresponding  to  this  time.  Thus, 
from  Table  XVIII  take  out  the  longitude  answering  to  the  given 
year,  and  from  Tables  XIX,  XX,  and  XXI,  take  out  the  motions 
in  longitude  for  the  given  month,  days,  hours,  and  minutes,  neg- 
lecting the  seconds.  The  sum  of  the  quantities  taken  from  the 
tables,  rejecting  12  signs,  when  it  exceeds  that  quantity,  will  be 
the  sun's  mean  longitude  for  the  given  time. 

With  the  sun's  mean  longitude  thus  found,  take  the  Equation 
of  Time  from  Table  XII,  Then,  when  Apparent  Time  is  given 
to  find  the  Mean,  apply  the  equation  with  the  sign  it  has  in  the 
table  ;  but  when  Mean  Time  is  given  to  find  the  Apparent,  apply 
it  with  the  contrary  sign  ;  the  result  will  be  the  Mean  or  Apparent 
Time  required. 

This  rule  will  be  sufficiently  exact  for  ordinary  purposes,  for 
several  years  before  and  after  the  year  1840.  When  the  given 
date  is  a  number  of  years  distant  from  this  epoch,  take  also  with 
the  sun's  mean  longitude  the  Secular  Variation  of  the  Equation  of 
Time  from  Table  XIII,  and  find  by  simple  proportion  the  variation 
in  the  interval  between  the  given  year  and  1840.  The  result,  ap- 
plied to  the  equation  of  time  taken  from  Table  XII,  according  to 
its  sign,  if  the  given  time  is  subsequent  to  the  year  1840,  but  with 
the  opposite  sign  if  it  is  prior  to  1.840,  will  give  the  equation  of 
time  at  the  given  date,  which  apply  to  the  given  time  as  above 
directed. 

Note  1.  When  the  exact  mean  or  apparent  time  to  within  a 
small  fraction  of  a  second  is  demanded,  take  the  numbers  in  the 
columns  entitled  I,  II,  III,  IV,  V,  N,  in  Tables,  XVIII,  XIX, 
XX,  answering  respectively  to  the  year,  month,  days,  and  hours, 
of  the  given  time.  With  the  respective  sums  of  the  numbers 
taken  from  each  column,  as  arguments,  enter  Table  XIV,  and 
take  out  the  corresponding  quantities.  These  quantities  added  to 
the  equation  of  time  as  given  by  Tables  XII  and  XIII,  and  the 


276  ASTRONOMICAL  PROBLEMS. 

constant  3.0s.  subtracted,  will  give  the  true  Equation  of  Time,  if 
the  given  time  is  Mean  Time.  When  Apparent  Time  is  given,  it 
will  be  farther  necessary  to  correct  the  equation  of  time  as  given 
by  the  tables,  by  stating  the  proportion,  24  hours  :  change  of 
equation  for  1°  of  longitude  :  :  equation  of  time  :  correction. 

Note  2.  The  Equation  of  Time  is  given  in  the  Nautical  Alma- 
nac for  each  day  of  the  year,  at  apparent,  and  also  at  mean  noon, 
on  the  meridian  of  Greenwich,  and  can  easily  be  found  for  any 
intermediate  time  by  a  proportion.  Directions  for  applying  it  to  the 
given  time  are  placed  at  the  head  of  the  column.  The  Equation 
is  given  on  the  first  and  second  pages  of  each  month. 

Exam.  1.  On  the  16th  of  July,  1840,  when  it  is  9h.  35m.  22s. 
P.  M.,  mean  time  at  New  York,  what  is  the  apparent  time  at  the 
same  place  ? 

Time  at  New  York,  July,  1840,      16d-  9h-  35m-  22s- 
Diff.  of  Long.         ...  4    56      4 

Time  at  Greenwich,  July,  1840,     16  14    31     26 

M.  Long. 

1840       .J      .         .         .         .         9s-  10°  12'  49" 
July    L'L      .         .         .         .         5    29    23   16 
16d.      '.  ,      ....  14   47     5 

•  I4h.      .       .     ..;  ::  *';..'..      .  34  30 

31m.  1    16 


M.  Long.       .  .         .         3    24   58  56 

The  equation  of  time  in  Table  XII,  corresponding  to  3'-  24°  58' 
56",  is  +  5m-  44s- 

Mean  Time  at  New  York,  July,  1840,  16d-  9h-  35m-  22s- 
Equation  of  time,  sign  changed,         .  — 5     44 

Apparent  Time,       .         .         .         .  16    9    29     38P.M. 

2.  On  the  9th  of  May,  1842,  when  it  is  4h.  15m.  21sec.  A.  M. 

apparent  time  at  New  Y  ork,  what  is  the  mean  time  at  the  same 

place,  and  also  at  Greenwich  ? 

Time  at  New  York,  May,  1842,     8d-  16h-  15m-  21'- 
Diff.  of  Long.         .  4     56      4 

Time  at  Greenwich,       .        ^ ,      8   21     11     25 

M.  Long. 

1842      .     -V         9*  10°  43' 18" 
May      .        .        3    28   16  40 
8d.  6   53  58 

21h.       .  51  45 

llm.  27 


M.  Long       >;-     1    16  46     8.  Equa.  of  time, — 3m.  45s, 


TO  CONVERT  APPARENT  INTO  MEAN  TIME. 


277 


Apparent  Time  at  Greenwich,  May,  1842,        8d-  21h-  llm-  25' 
Equation  of  Time,       *-.         . 


-3     45 


Mean  Time  at  Greenwich, 
Diff.  of  Long.        .         « 


8     21      7    40 
4    56       4 

8     16    11     36 


Mean  Time  at  New  York, 

Or,  May  9th,  4h.  llm.  36s.  A.  M. 

3.  On  the  3d  of  February,  1855,  when  it  is  2h,  43m.  36s.  appa- 
rent time  at  Greenwich,  what  is  the  exact  mean  time  at  the  same 
place  ? 

Appar.  Time  at  Greenwich,  Feb.,  1855,  3d.  2h.  43m.  36s. 


M,  Long. 

I. 

II. 

III. 

IV. 

V. 

N. 

1855 

9'  10°  34'  30" 

433 

279 

806 

889 

866 

863 

Feb. 

1   0  33  18 

47 

85 

138 

45 

7 

5 

3d. 

1  58  17 

68 

5 

9 

3 

0 

0 

2h. 

4  56 

3 

43m. 

1  46 

10  13  12  47 

551 

369 

953 

937 

873 

868 

Appar.  Time  at  Greenwich,  Feb.,  1855,  3d-  2h-  43m-  36s- 
Equation  of  time  by  Table  XII,  .  +14       8.6 

lOOyrs.  :  13s.  (Sec.  Var.,  Table  XIII) 

:  :  15yrs. :  1.9s.  .         .         .  —1.9 


Approx.  Mean  Time  at  Greenwich, 
24h. :  6s.  (change  of  equa.  for  T 
long.) : :  14m. :  O.ls. 

II.  Ill 

II.  IV.       .... 

II.  V 

I 

N 

Constant. 


of 


3    2    57     42.7 

+0.1 
0.8 
0.4 
1.0 
0.3 
0.1 

—3.0 


Mean  Time  at  Greenwich,  3    2    57     42.4 

4.  On  the  18th  of  November,  1841,  when  it  is  2h.  12m.  26sec. 
A.  M.  mean  time  at  Greenwich,  what  is  the  apparent  time  at 
Philadelphia?  Ans.  Nov.  17th,  9h.  26m.  28s.  P.  M. 

5.  On  the  2d  of  February,  1839,  when  it  is  6h.  32m.  35sec. 
P.  M.,  apparent  time  at  New  Haven,  what  is  the  mean  time  at  the 
same  place  ?  Ans.  6h.  46m.  39s.  P.  M. 

6.  On  the  23d  of  September,  1850,  when  it  is  9h.  10m.  12sec. 
mean  time  at  Boston,  what  is  the  exact  apparent  time  at  the  same 
place?  Ans.  9h.  18m.  1.0s. 


278  ASTRONOMICAL  PROBLEMS. 


PROBLEM  VII. 

To  correct  the  Observed  Altitude  of  a  Heavenly  Body  for  Re  - 

fraction. 

With  the  given  altitude  take  the  corresponding  refraction  from 
Table  VIII.  Subtract  the  refraction  from  the  given  altitude,  and 
the  result  will  be  the  true  altitude  of  the  body  at  the  given  station. 

This  rule  will  give  exact  results  if  the  barometer  stands  at  30 
inches,  and  Fahrenheit's  thermometer  at  50°,  and  results  suffi- 
ciently exact  for  ordinary  purposes  in  any  state  of  the  atmosphere. 
When  there  is  occasion  for  greater  precision,  take  from  Table  IX 
the  corrections  for  +  1  inch  in  the  height  of  the  barometer,  and 
—  1°  in  the  height  of  Fahrenheit's  thermometer,  and  compute  the 
corrections  for  the  difference  between  the  observed  height  of  the 
barometer  and  30in.  and  for  the  difference  between  the  observed 
height  of  the  thermometer  and  50°.  Add  these  to  the  mean  re- 
fraction taken  from  Table  VIII,  if  the  barometer  stands  higher 
than  30in.  and  the  thermometer  lower  than  50°  ;  but  in  the  oppo- 
site case  subtract  them,  and  the  result  will  be  the  true  refraction, 
which  subtract  from  the  observed  altitude. 

Exam.  1.  The  observed  altitude  of  the  sun  being  32°  10'  25", 
what  is  its  true  altitude  at  the  place  of  observation  ? 

Observed  alt.         .         .         .         32°  10'  25" 
Refraction  (Table  VIII)         .  —1  32 

True  alt.  at  the  station,  .         32°    8  53 

2.  The  observed  altitude  of  Sirius  being  20°  42'  11",  the  ba- 
rometer 29.5  inches,  and  the  thermometer  of  Fahrenheit  70% 
required  the  true  altitude  at  the  place  of  observation.  The  differ- 
ence between  29.5  inches  and  30  inches  is  0.5  inches,  and  the 
difference  between  70°  and  50°  is  20°. 
Obs.  alt.  20°42'11".0 


Refrac. (Table VIII),  2'  33".0;  Bar.+lin.,5".12;ther.-l°.0".310 
Corr.for-0.5in.,bar.  -2  .6  .5  20 

Corr.for+20°,ther.      —6  .2 

2.560  6.20 

True  refrac.  2  24  .2 


True  alt.  20  39  46  .8 

3.  The  observed  altitude  of  the  moon  on  the  llth  of  April,  1838, 
being  14°  17'  20",  required  the  true  altitude  at  the  place  of  obser- 
vation. Ans.  14°  13'  35''. 

4.  Let  the  observed  altitude  of  Aldebaran  be  48°  35'  52",  the 
barometer  at  the  same  time  standing  at  30.7  inches,  and  the  ther- 
mometer at  42°,  required  the  true  altitude.   Ans.  48°  34'  58".8. 


* 

TO  DEDUCE  THE  TRUE  FROM  THE  APPARENT  ALTITUDE.   279 

PROBLEM  VIII. 

The  Apparent  Altitude  of  a  Heavenly  Body  being  given,  to  find 
its  True  Altitude. 

Correct  the  observed  altitude  for  refraction  by  the  foregoing 
problem.  Then, 

1.  If  the  sun  is  the  body  whose  altitude  is  taken,  find  its  paral- 
lax in  altitude  by  Table  X,  and  add  it  to  the  observed  altitude  cor- 
rected for  refraction.     The  result  will  be  the  true  altitude  sought. 

2.  If  it  is  the  altitude  of  the  moon  that  is  taken,  and  the  hori- 
zontal parallax  at  the  time  of  the  observation  is  known,  find  the 
parallax  in  altitude  by  the  following  formula : 

log.  sin  (par.  in  alt.)  =  log.  sin  (hor.par.)  -Hog.  cos  (app.alt.)  — 10 ; 

and  add  it,  as  before,  to  the  apparent  altitude  corrected  for  refrac- 
tion. 

3.  If  one  of  the  planets  is  the  body  observed,  the  following  for- 
mula will  serve  for  the  determination  of  the  parallax  in  altitude 
when  the  horizontal  parallax  is  known : 

log.  (par.  in  alt.)  =  log.  (hor.  par.)  +  log.  cos  (appar.  alt) — 10. 

Note  1 .  The  equatorial  horizontal  parallax  of  the  moon  at  any 
given  time  may  be  obtained  from  the  tables  appended  to  the  work. 
(See  Problem  XIV.)  But  it  can  be  had  much  more  readily  from 
the  Nautical  Almanac.  The  equatorial  horizontal  parallax  being 
known,  the  horizontal  parallax  at  any  given  latitude  may  be  ob- 
tained by  subtracting  the  Reduction  of  Parallax,  to  be  found  in 
Table  LXIV.  The  horizontal  parallax  of  any  planet,  the  altitude 
of  which  is  measured,  may  also  be  derived  from  the  Nautical  Al- 
manac. 

Note  2.  The  fixed  stars  have  no  sensible  parallax,  and  thus  the 
observed  altitude  of  a  star,  corrected  for  refraction,  will  be  its  true 
altitude  at  the  centre  of  the  earth  as  well  as  at  the  station  of  the 
observer. 

Note  3.  If  the  true  altitude  of  a  heavenly  body  is  given,  and  it 
is  required  to  find  the  apparent,  the  rules  for  finding  the  parallax 
in  altitude  and  the  refraction  are  the  same  as  when  the  apparent 
altitude  is  given  ;  the  true  altitude  being  used  in  place  of  the  ap- 
parent. But  these  corrections  are  to  be  applied  with  the  opposite 
signs  from  those  used  in  the  determination  of  the  true  altitude  from 
the  apparent ;  that  is,  the  parallax  is  to  be  subtracted,  and  the  re- 
fraction added.  It  wil]  also  be  more  accurate  to  make  use  of 
equa.  (10),  p.  52,  in  the  case  of  the  moon. 

Exam.  1.  The  observed  altitude  of  the  sun  on  the  1st  of  May 
1837,  being  26°  40'  20",  what  is  its  true  altitude  ? 


280 


ASTRONOMICAL  PROBLEMS. 


Obs.  alt. 

Refraction     .     \-^9  r     . 

True  alt.  at  the  station, 
Parallax  in  alt.  (Table  X), 


26J  40'  20" 
-1  56 

26    38  24 

+  8 

26    38  32 


True  altitude 

2.  Let  the  apparent  altitude  of  the  moon  at  New  York  on  the 
17th  of  March,  1837,  8h.  P.  M.,  be  66°  10'  44" ;  the  barometer 
30.4in.  and  the  thermometer  62°  ;  required  the  true  altitude. 
Appar.  alt.  .         .         66°  10'  44" 

Meanrefrac.         .  0  25.7 

Corr.  for  +  0.4in.,  bar.  +  0.3 

Corr.  for  +  12°,  ther.  —0.6 


True  refrac. 


0  25.4 


True  alt.  at  N.  York,    66  10  18.6 
Equa.  par.  by  N.  Almanac,  54'  13" 
Reduc.  for  lat.  40°,  4 


Hor.  par.  at  New  York,       54     9 
Par.  in  alt. 


logarithms, 
cos.  9.60637 


sin.  8.19731 


21  52 


sin.  7.80368 


True  altitude     .         .  66  32  11 

3.  On  the  18th  of  February,  1837,  the  true  meridian  altitude  of 
the  planet  Jupiter  at  Greenwich  was  56°  54'  57",  what  was  its 
apparent  altitude  at  the  time  of  the  meridian  passage,  the  horizontal 
parallax  being  taken  at  1".9,  as  given  by  the  Nautical  Almanac  ? 

True  alt.  .         .         56°  54'  57';     .     cos.  9.7371 

Hor.  par.  1".9  .         .         .         .      ....        log.  0.2787 


Par.  in  alt. 
Refraction 


—1.0 
+  37.9 


log.  0.0158 


Appar.  alt.         .         .         56    55  34 

4.  What  will  be  the  true  altitude  of  the  sun  on  the  22d  of  Sep- 
tember, 1840,  at  the  time  its  apparent  altitude  is  39°  17'  50"  ? 

Ans.  39°  16'  46". 

5.  Given  29°  33'  30"  the  apparent  altitude  of  the  moon  at  Phil 
adelphia  on  the  15th  of  June,  1837,  at  9h.  30m.  P.  M.,  and  58'  33' 
the  equatorial  parallax  of  the  moon  at  the  same  time,  to  find  tht 
true  altitude.  Ans.  30°  22'  41". 

6.  Given  15°  24'  23"  the  true  altitude  of  Venus,  and  8"  its  hori- 
zontal parallax,  to  find  the  apparent  altitude    Ans.  15°  27'  41". 


TO  FIND  THE  SUN*S  LONGITUDE,  ETC  ,  FROM  TABLES.         281 


PROBLEM  IX. 

To  find  the  Sun's  Longitude,  Hourly  Motion,  and  Semi-diameter, 
for  a  given  time,  from  the  Tables. 

For  the  Longitude. 

When  the  given  time  is  not  for  the  meridian  of  Greenwich,  re- 
duce it  to  that  meridian  by  Problem  V  ;  and  when  it  is  apparent 
time,  convert  it  into  mean  time  by  the  last  problem. 

With  the  mean  time  at  Greenwich,  take  from  Tables  XVIII, 
XIX,  XX,  and  XXI,  the  quantities  corresponding  to  the  year, 
month,  day,  hour,  minute,  and  second,  (omitting  those  in  the  last 
two  columns,)  and  place  them  in  separate  columns  headed  as  in 
Table  XVIII,  and  take  their  sums.*  The  sum  in  the  column  enti- 
tled M.  Long,  will  be  the  tabular  mean  longitude  of  the  sun ;  the 
sum  in  the  column  entitled  Long.  Perigee  will  be  the  tabular  lon- 
gitude of  the  sun's  perigee ;  and  the  sums  in  the  columns  I,  II, 
III,  IV,  V,  N,  will  be  the  arguments  for  the  small  equations  of  the 
sun's  longitude,  including  the  equation  of  the  equinoxes  in  longi- 
tude. 

Subtract  the  longitude  of  the  perigee  from  the  sun's  mean  longi- 
tude, adding  12  signs  when  necessary  to  render  the  subtraction 
possible ;  the  remainder  will  be  the  sun's  mean  anomaly.  With 
the  mean  anomaly  take  the  equation  of  the  sun's  centre  from  Ta- 
ble XXV,  and  correct  it  by  estimation  for  the  proportional  part  of 
the  secular  variation  in  the  interval  between  the  given  year  and 
1840;  also  with  the  arguments  I,  II,  III,  IV,  V,  take  the  corre- 
sponding equations  from  Tables  XXVIII,  XXX,  XXXI,  and 
XXXII.  The  equation  of  the  centre  and  the  four  other  equations, 
together  with  the  constant  3",  added  to  the  mean  longitude,  will 
give  the  sun's  True  Longitude,  reckoned  from  the  Mean  Equinox. 

With  the  argument  N  take  the  equation  of  the  equinoxes  or  Lu- 
nar Nutation  in  Longitude  from  Table  XXVII.  Also  take  the  So- 
lar Nutation  in  longitude,  answering  to  the  given  date,  from  the 
same  table.  Apply  these  equations  according  to  their  signs  to  the 
true  longitude  from  the  mean  equinox,  already  found ;  the  result 
will  be  the  True  Longitude  from  the  Apparent  Equinox. 

For  the  Semi-diameter  and  Hourly  Motion. 

With  the  sun's  mean  anomaly,  take  the  hourly  motion  and  semi- 
diameter  from  Tables  XXIII  and  XXIV. 

*  In  adding  quantities  that  are  expressed  in  signs,  degrees,  &c.,  reject  12  or  24 
signs  whenever  the  sum  exceeds  either  of  these  quantities.  In  adding  arguments 
expressed  in  100  or  1000,  &c.  parts  of  the  circle,  when  they  consist  of  two  figures, 
reject  the  hundreds  from  the  sum;  when  of  three  figures,  the  thousands;  antf 
when  of  four  figures,  the  ten  thousands. 

36 


282 


ASTRONOMICAL  PROBLEMS. 


Notes. 

1 .  If  the  tenths  of  seconds  be  omitted  in  taking  the  equations 
from  the  tables  of  double  entry,  the  error  cannot  exceed  2" ;  ir 
case  the  precaution  is  taken  to  add  a  unit,  whenever  the  tenths  ex- 
ceed .5. 

2.  The  longitude  of  the  sun,  obtained  by  the  foregoing  rule, 
may  differ  about  3"  from  the  same  as  derived  from  the  most  accu- 
rate solar  tables  now  in  use.     When  there  is  occasion  for  greater 
precision,  take  from  Tables  XVIII,  XIX,  and  XX,  the  quantities 
in  the  columns  entitled  VI  and  VII,  along  with  those  in  the  other 
columns.    With  the  sums  in  these  columns,  and  those  in  the  col- 
umns I,  II,  as  arguments,  take  the  corresponding  equations  from 
Tables  XXIX  and  XXXIII.     Also  with  the  sun's  mean  anomaly 
take  the  equation  for  the  variable  part  of  the  aberration  from  Ta- 
ble XXXIV.     Add  these  three  equations  along  with  the  others  to 
the  mean  longitude,  and  omit  the  addition  of  the  constant  3".    The 
result  will  be  exact  to  within  a  fraction  of  a  second. 

Exam.  1 .  Required  the  sun's  longitude,  hourly  motion,  and  se- 
mi-diameter, on  the  25th  October,  1837,  at  llh.  27m.  38s.  A.  M 
mean  time  at  New  York. 

Mean  time  at  N.  York,  Oct.  1837,  24d-  23h-  27m-  38s- 
Diif.  of  Long 4     56        4 


Mean  time  at  Greenwich, 


25     4     23      42 


1837    . 

October 
25d.  . 
4h.  . 
23m.  . 
42s. 


Eq.  Sun's  Cent. 

II.  III. 
II.  IV. 
II.  V. 

Const.  . 


Lunar  Nutation 
Solar  Nutation 

Sun'strue  long. 


M.  Long. 


9  10  55  47.2 

8  29     4  54.1 

23  39  19,9 

9  51.4 

56.7 

1.7 


7     3  50  51.0 

11  28  12  43.5 

2.5 

9.0 

7.7 

19.3 

3.0 


2     4  16.0 

—  6.3 

—  1.2 


7     2    4    8.5 


Long.  Perigee.  I. 


II. 


9  10     8     5816280 


46250748215 
4810J  66  107 

8      0 


9  10     8  55882    94 
7     3  50  51 


III.  IV. 


549 


872 


321 

397 

35 


753 


v.  IN. 

348895 


63 


40 


416  939 


9  23  41  56  Mean  Anomaly. 
Sun's  Hourly  Motion,      .     .  2'  29". 7 
Sun's  Semi-diameter  16'  17".2 


2.  Required  the  sun's  longitude,  hourly  motion,  and  semi-diam 
eter,  on  the  15th  of  July,  1837,  at  8h.  20m.  40s.  P.  M.  mean  time 
at  Greenwich. 


TO  FIND  THE  APPARENT  OBLIQUITY  OF  THE  ECLIPTIC. 


1837 
July 
15d. 

8h. 
20m. 
40s.  . 

Eq.  Sun's  Cent. 

ii.  in! 

II.    IV. 
II.     V. 
I.   VI. 
II.  VII. 

Aber.     . 

Lunar  Nutation 
Solar  Nutation 

Sun's  true  long. 

M.  Long. 

Long.  Peri. 

I. 

816 
129 
473 
11 

II.  I  III 

IV.  V. 

N.  VI. 

VII. 

so/         // 

9  10  55  47.2 
5  28  24     7.8 
13  47  56.6 
19  42.8 
49.3 
1.6 

8        0          / 

9  10     8     5 
31 

2 

280549 
496,806 

38   62 
1      1 

321  348 
263  41 
20     3 

604392 

895  787 
27569 
2508 
11 

600 
17 
2 

9  10    8  38 
3  23  28  25 

429l8is|418 

924875   619 

2'  23".l 
.      15'  45".4 

3  23  28  25.3 
11  29  33  10.3 
10.7 
6.6 
5.0 
7.7 
1.8 
0.2 
0.6 

6  13  19  47  Mean  Anomaly. 
Sun's  Hourly  Motion, 
Sun's  Semi-diameter, 

3  23    2    8.2 

—  7.8 
+  0.8 

3  23    2     1.2 

3.  Required  the  sun's  longitude,  hourly  motion,  and  semi-diam 
eter,  on  the  10th  of  June,  1838,  at  9h.  45m.  26s.  A.  M.  mean  time 
at  Philadelphia,  (omitting  the  three  smallest  equations  of  long* 
tude.) 

Ans.    Sun's  longitude,  2s- 19°  11'  57"  ;  hourly  motion,  2'  23".3  ; 
semi-diameter,  15'  46". 1. 

4.  Required  the  sun's  longitude,  hourly  motion,  and  semi-diam< 
eter,  on  the  1st  of  February,  1837,  at  12h.  30m.  15s.  mean  astro- 
nomical time  at  Greenwich. 

Ans.     Sun's  longitude,  10s-  13°  I7   44". 6 ;  hourly  motion,  2f 
32".l ;  semi-diameter,  16'  14".7. 


PROBLEM  X. 


To  find  the  Apparent  Obliquity  of  the  Ecliptic,  for  a  given  time, 
from  the  Tables. 

Take  the  mean  obliquity  for  the  given  year  from  Table  XXII. 
Then  with  the  argument  N,  found  as  in  the  foregoing  problem, 
and  the  given  date,  take  from  Table  XXVII  the  lunar  and  solar 
nutations  of  obliquity.  Apply  these  according  to  their  signs  to  the 
mean  obliguity,  and  the  result  will  be  the  apparent  obliquity. 

Exam.  1 .  Required  the  apparent  obliquity  of  the  ecliptic  on  the 
15th  of  March,  1839. 


284  ASTRONOMICAL  PROBLEMS. 

N. 

1839,  .  3 
March,  9 
15d.  .  2 

M.  Obliquity,       23°  27'  36". 9 

14          ...  +9  .1 

Solar  Nutation  for  March  15th,  +0  .5 


Apparent  Obliquity,     .         .         23  27  46  .5 
2.  Required  the  apparent  obliquity  of  the  ecliptic  on  the  12th 
of  July,  1845.  Ans.  23°  27'  28". 2. 


PROBLEM  XL 

Given  the  Sun's  Longitude  and  the  Obliquity  of  the  Ecliptic,  tc 
find  his  Right  Ascension  and  Declination* 

Let  w  =  obliquity  of  the  ecliptic  ;  L  =  sun's  longitude  ;  R  == 
sun's  right  ascension ;  and  D  =  sun's  declination ;  then  to  find  R 
and  D,  we  have 

log.  tang  R  =  log.  tang  L  +  log.  cos  u  —  10, 
log.  sin  D  =  log.  sin  L  +  log.  sin  w  —  10. 

The  right  ascension  must  always  be  taken  in  the  same  quadrant 
as  the  longitude.  The  declination  must  be  taken  less  than  90°  ; 
and  it  will  be  north  or  south  according  as  its  trigonometrical  sine 
comes  out  positive  or  negative. 

Note.  The  sun's  right  ascension  and  declination  are  given  in 
the  Nautical  Almanac  for  each  day  in  the  year  at  noon  on  the  me- 
ridian of  Greenwich,  and  may  be  found  at  any  intermediate  time 
by  a  proportion. 

Exam.  1.  Given  the  sun's  longitude  205°  23'  50",  and  the  ob- 
:iquity  of  the  ecliptic  23°  27'  36",  to  find  his  right  ascension  and 
declination. 

L=205°  23'    50"       ...         tan.   9.67649 
w  =    23     27     36  cos.   9.96253 


R  =  203     32       5  tan.   9.63902 


L  =  205     23     50         .         .         .         sin.    9.63235— 
w  =    23     27     36         .     '    .         .         sin.    9.60000 

D=      9     49     52  S.    .       '.--     .         sin.    9.23235  — 
2.  The  obliquity  of  the  ecliptic  being  23°  27'  30",  required 

*  The  obliquity  of  the  ecliptic  at  any  given  time  for  which  the  sun's  longitude 
is  known,  is  found  by  the  foregoing  Problem. 


TO  FIND  THE  SUN  S  LONGITUDE  AND  DECLINATION.  285 

the  sun's  right  ascension  and  decimation  when  his  longitude  if 
44°  18'  25". 

Aiis.  Right  ascension  41°  50'  30",  and  declination  16°  8'  40"  N. 


PROBLEM  XII. 

Given  the  Surfs  Right  Ascension  and  the  Obliquity  of  the  Eclip 
tic,  to  find  his  Longitude  and  Declination. 

Using  the  same  notation  as  in  the  last  problem,  we  have,  to  find 
the  longitude  and  declination, 

log.  tang  L  =  log.  tang  R  +  ar.  co.  log.  cos  w, 
log.  tang  D  =  log.  sin  R  +  log.  tang  w  —  10. 

Exam.  1.  What  is  the  longitude  and  declination  of  the  sun, 
when  his  right  ascension  is  142°  11'  34",  and  the  obliquity  of  the 
ecliptic  23°  27'  40"  ? 

R  =  142°    II7    34"       .     /Vv      .         tan.    9.88979  — 

w  =   23     27     40    '-...i'*.  •.',...    ar.  co.  cos.    0.03747 

L  =  139     46     30         .         .         .         tan.     9.92726  — 

R=142     11     34         .     r.v       .         sin.    9.78746 
w=    23     27     40         .         .         .         tan.    9.63750 

D=    14     53     55  N.    .         .      ..,;.,-     tan.    9.42496 

2.  Given  the  sun's  right  ascension  310°  25'  11",  and  the  obli- 
quity of  the  ecliptic  23°  27'  35",  to  find  the  longitude  and  declina- 
tion. 

Ans.  Longitude  307°  59'  57",  and  declination  18°  17'  0"  S. 


PROBLEM   XIII. 

The  Surfs  Longitude  and  the  Obliquity  of  the  Ecliptic  being 
given,  to  find  the  Angle  of  Position. 

Let  p  ~  angle  of  position  ;  w  =  obliquity  of  the  ecliptic ;  and 
L  =  sun's  longitude.  Then, 

log.  tangp  =  log.  cos  L  +  log.  tang  w  —  10. 

The  angle  of  position  is  always  less  than  90°.  The  northern 
part  of  the  circle  of  latitude' will  lie  on  the  west  or  east  side  of  the 
northern  part  of  the  circle  of  declination,  according  as  the  sign  of 
the  tangent  of  the  angle  of  position  is  positive  or  negative. 

Exam.  1.  Given  the  sun's  longitude  24°  15'  20",  and  the  obli 
quity  of  the  ecliptic  23°  27'  32",  required  the  angle  of  position. 


ASTRONOMICAL  PROBLEMS. 

L  =  24°    15'    20"       .         .         cos     9.9598G 
w  =  23     27     32         .         .         tan.    9.63745 

p=2l     35     10         .         .         tan.    9.59731 

The  northern  part  of  the  circle  of  latitude  is  to  the  west  of  the 
circle  of  declination. 

2.  When  the  sun's  longitude  is  120°  18'  55",  and  the  obliquity 
of  the  ecliptic  23°  27'  30",  what  is  the  angle  of  position  ? 

Ans.  12°  21'  17"  ;  and  the  northern  part  of  the  circle  of  latitude 
lies  to  the  east  of  the  circle  of  declination. 


PROBLEM  XIV. 

To  find  from  the  Tables,  the  Moon's  Longitude,  Latitude,  Equa- 
torial Parallax,  Semi-diameter,  and  Hourly  Motion  in  Longi- 
tude and  Latitude,  for  a  given  time. 

When  the  given  time  is  not  for  the  meridian  of  Greenwich,  re- 
duce it  to  that  meridian,  and  when  it  is  apparent  time  convert  it 
into  mean  time. 

Take  from  Table  XXXV,  and  the  following  tables,  the  argu- 
ments numbered  1,  2,  3,  &c.,  to  20,  for  the  given  year,  and  their 
variations  for  the  given  month,  days,  &c.,  and  find  the  sums  of  the 
numbers  for  the  different  arguments  respectively ;  rejecting  the 
hundred  thousands  and  also  the  units  in  the  first,  the  ten  thousands 
in  the  next  eight,  and  the  thousands  in  the  others.  The  resulting 
quantities  will  be  the  arguments  for  the  first  twenty  equations  of 
longitude. 

With  the  same  time,  take  from  the  same  tables  the  remaining 
arguments  with  their  variations,  entitled  Evection,  Anomaly,  Va- 
riation, Longitude,  Supplement  of  the  Node,  II,  V,  VI,  VII,  VIII, 
IX,  and  X ;  and  add  the  quantities  in  the  column  for  the  Supple- 
ment of  the  Node. 

For  the  Longitude. 

With  the  first  twenty  arguments  of  longitude,  take  from  Tables 
XLI  to  XLVI,  inclusive,  the  corresponding  equations  ;  and  with 
the  Supplement  of  the  Node  for  another  argument,  take  the  corre- 
sponding equation  from  Table  XLIX.  Place  these  twenty-one 
equations  in  a  single  column,  entitled  Eqs.  of  Long. ;  and  write 
beneath  them  the  constant  55".  Find  the  sum  of  the  whole,  and 
place  it  in  the  column  of  Evection.  Then  the  sum  of  the  quanti- 
ties in  this  column  will  be  the  corrected  argument  of  Evection. 

With  the  corrected  argument  of  Evection,  take  the  Evection 
from  Table  L,  and  add  it  to  the  sum  in  the  column  of  Eqs.  of 
Long.  Place  this  in  the  column  of  Anomaly.  Then  the  sum  of 
the  quantities  in  this  column  will  be  the  corrected  Anomaly. 


TO  FIND  THE  MOON'S  LONGITUDE,  ETC.  287 

With  the  corrected  Anomaly,  take  the  Equation  of  the  Centre 
from  Table  LI,  and  add  it  to  the  last  sum  in  the  column  of  Eqs. 
of  Long.  Place  the  resulting  sum  in  the  column  of  Variation. 
Then  the  sum  of  the  quantities  in  this  column  will  be  the  corrected 
argument  of  Variation. 

With  the  corrected  argument  of  Variation,  take  the  variation 
from  Table  LII,  and  add  it  to  the  last  sum  in  the  column  of  Eqs. 
of  Long. ;  the  result  will  be  the  sum  of  the  principal  equations 
of  the  Orbit  Longitude,  amounting  in  all  to  twenty-four,  and  the 
constants  subtracted  for  the  other  equations.  Place  this  sum  in 
the  column  of  Longitude.  Then  the  sum  of  the  quantities  in  this 
column  will  be  the  Orbit  Longitude  of  the  Moon,  reckoned  from 
the  mean  equinox. 

Add  the  orbit  longitude  to  the  supplement  of  the  node,  and  the 
resulting  sum  will  be  the  argument  of  Reduction. 

With  the  argument  of  Reduction,  take  the  Reduction  from  Ta- 
ble LIII,  and  add  it  to  the  Orbit  Longitude.  The  sum  will  be  the 
Longitude  as  reckoned  from  the  mean  equinox.  With  the  Supple- 
ment of  the  Node,  take  the  Nutation  in  Longitude  from  Table 
LI V,  and  apply  it,  according  to  its  sign,  to  the  longitude  from-  the 
mean  equinox.  The  result  will  be  the  Moon's  True  Longitude 
from  the  Apparent  Equinox. 

For  the  Latitude. 

The  argument  of  the  Reduction  is  also  the  1st  argument  of  Lat- 
itude. Place  the  sum  of  the  first  twenty-four  equations  of  Longi- 
tude, taken  to  the  nearest  minute,  in  the  column  of  Arg.  II.  Find 
the  sum  of  the  quantities  in  this  column,  and  it  will  be  the  Arg.  II 
of  Latitude,  corrected.  The  Moon's  true  Longitude  is  the  3d  ar- 
gument of  Latitude.  The  20th  argument  of  Longitude  is  the  4th 
argument  of  Latitude.  Take  from  Table  LVIII  the  thousandth 
parts  of  the  circle,  answering  to  the  degrees  and  minutes  in  the 
sum  of  the  first  twenty-four  equations  of  longitude,  and  place  it  in 
the  columns  V,  VI,  VII,  VIII,  and  IX  ;  but  not  in  the  column  X. 
Then  the  sums  of  the  quantities  in  columns  V,  VI,  VII,  VIII,  IX, 
and  X,  rejecting  the  thousands,  will  be  the  5th,  6th,  7th,  8th,  9th, 
and  10th  arguments  of  Latitude. 

With  the  Arg.  I  of  Latitude,  take  the  moon's  distance  from  the 
North  Pole  of  the  Ecliptic,  from  Table  LV  ;  .and  with  the  remain- 
ing nine  arguments  of  latitude,  take  the  corresponding  equations 
from  Tables  LVI,  LVII,  and  LIX.  The  sum  of  these  quantities, 
increased  by  10",  will  be  the  moon's  true  distance  from  the  North 
Pole  of  the  Ecliptic.  The  difference  between  this  distance  and 
90°  will  be  the  Moon's  true  Latitude ;  which  will  be  North  or 
South)  according  as  the  distance  is  less  or  greater  than  90°. 

For  the  Equatorial  Parallax. 
With  the  corrected  arguments,  Evection,  Anomaly,  and  Varia 


288  ASTRONOMICAL  PROBLEMS. 

tion,  take  out  the  corresponding  quantities  from  Tables  LXI, 
LXII,  and  LXIII.  Their  sum,  increased  by  7",  will  be  the  Equa- 
torial Parallax. 

For  the  Semi-diameter. 

With  the  Equatorial  Parallax  as  an  argument,  take  out  the 
moon's  semi-diameter  from  Table  LXV. 

For  the  Hourly  Motion  in  Longitude. 

With  the  arguments  2,  3,  4,  5,  and  6  of  Longitude,  rejecting  the 
two  right-hand  figures  in  each,  take  the  corresponding  equations 
of  the  hourly  motion  in  longitude  from  Table  LXVII.  Find  the 
sum  of  these  equations  and  the  constant  3",  and  with  this  sum  at 
the  top,  and  the  corrected  argument  of  the  Evection  at  the  side, 
take  the  corresponding  equation  from  Table  LXIX  ;  also  with  the 
corrected  argument  of  the  Evection  take  the  corresponding  equa- 
tion from  Table  LXVIII. 

Add  these  equations  to  the  sum  just  found,  and  with  the  result- 
ing sum  at  the  top,  and  the  corrected  anomaly  at  the  side,  take  the 
corresponding  equation  from  Table  LXX  ;  also  with  the  corrected 
anomaly  take  the  corresponding  equation  from  Table  LXXI. 

Add  these  two  equations  to  the  sum  last  found,  and  with  the  re- 
sulting sum  at  the  top,  and  the  corrected  argument  of  the  Variation 
at  the  side,  take  the  corresponding  equation  from  Table  LXXII. 
With  the  corrected  argument  of  the  Variation,  take  the  correspond- 
ing equation  from  Table  LXXIII. 

Add  these  two  equations  to  the  sum  last  found,  and  with  the  re- 
sulting sum  at  the  top,  and  the  argument  of  the  Reduction  at  the 
side,  take  the  corresponding  equation  from  Table  LXXIV.  Also, 
with  the  argument  of  the  Reduction  take  the  corresponding  equa- 
tion from  Table  LXXV.  These  two  equations,  added  to  the  last 
sum,  will  give  the  sum  of  the  principal  equations  of  the  hourly 
motion  in  longitude,  and  the  constants  subtracted  for  the  others. 
To  this  add  the  constant  27'  24".0,  and  the  result  will  be  the 
Moon's  Hourly  Motion  in  Longitude. 

For  the  Hourly  Motion  in  Latitude. 

With  the  argument  I  of  Latitude,  take  the  corresponding  equa- 
tion from  Table  LXXIX.  With  this  equation  at  the  side,  and  the 
sum  of  all  the  eouations^of  the  hourly  motion  in  longitude,  except 
the  last  two,  at  tne  top,  take  the  corresponding  equation  from  Ta- 
ble LXXXI.  With  the  argument  II  of.  Latitude,  take  the  corre- 
sponding equation  from  Table  LXXXII.  And  with  this  equation 
at  the  side,  and  the  sum  of  all  the  equations  of  the  hourly  motion 
in  longitude,  except  the  last  two,  at  the  top,  take  the  equation  from 
Table  LXXXIII.  Find  the  sum  of  these  four  equations  and  the 


TO  FIND  THE  MOON's  LONGITUDE,  ETC.  289 

constant  1".  To  the  resulting  sum  apply  the  constant  —  237".2. 
The  difference  will  be  the  Moon's  true  Hourly  Motion  in  Latitude. 
*The  moon  will  be  tending  North  or  South,  according  as  the  sign 
is  positive  or  negative. 

Note.  The  errors  of  the  results  obtained  by  the  foregoing  rules, 
occasioned  by  the  neglect  of  the  smaller  equations,  cannot  exceed 
for  the  longitude  15",  for  the  latitude  8",  for  the  parallax  7",  for 
the  hourly  motion  in  longitude  5",  and  for  the  hourly  motion  in 
latitude  3"  ;  and  they  will  generally  be  very  much  less.  When 
greater  accuracy  is  required,  take  from  Tables  XXXV  to  XXXIX 
the  arguments  from  21  to  31,  along  with  those  from  1  to  20,  and 
their  variations.  The  sums  of  the  numbers  for  these  different  ar- 
guments, respectively,  will  be  the  arguments  of  eleven  small  addi- 
tional equations  of  longitude.  Also,  take  from  the  same  tables  the 
arguments  entitled  XI  and  XII,  along  with  those  in  the  preceding 
columns.  Retain  the  right-hand  figure  of  the  sum  in  column  1  of 
arguments,  and  conceive  a  cipher  to  be  annexed  to  each  number 
in  the  columns  of  arguments  of  Table  XLI.  The  numbers  in  the 
columns  entitled  Diff.for  10,  will  then  be  the  differences  for  a  va- 
riation of  100  in  the  argument. 

For  the  Longitude.  With  the  arguments  21  to  31,  take  the  cor 
responding  equations  from  Tables  XLVII  and  XLVIII,  and  place 
them  in  the  same  column  with  the  equations  taken  out  with  the 
arguments  1,  2,  &c.  to  20.  Take  also  equation  32  from  Table 
XLIX,  as  before.  Find  the  sum  of  the  whole,  (omitting  the  con- 
stant 55",)  and  then  continue  on  as  above.  The  longitude  from 
the  mean  equinox  being  found,  take  the  lunar  nutation  in  longitude 
from  Table  LIV,  and  the  solar  nutation  answering  to  the  given 
date  from  Table  XXVII.  Apply  them  both,  according  to  their 
sign,  to  the  longitude  from  the  mean  equinox,  and  the  result  will 
be  the  more  exact  longitude  from  the  apparent  equinox,  required. 

For  the  Latitude.  With  the  arguments  XI  and  XII,  take  the 
corresponding  equations  from  Table  LIX.  Add  these  with  the 
other  equations,  and  omit  the  constant  10".  The  difference  beT 
tween  the  sum  and  90°  will  be  the  more  exact  latitude. 

For  the  Equatorial  Parallax.    With  the  arguments  1,  2,  4,  5, 

6,  8,  9,  12,  13,  take  the  corresponding  equations  from  Table  LX. 
Find  the  sum  of  these  and  the  other  equations,  omitting  the  con- 
stant 7",  and  it  will  be  the  more  exact  value  of  the  Parallax. 

For  the  Hourly  Motion  in  Longitude.     With  the  arguments  1, 

7,  8,  9,  10,  11,  12,  13,  14,  15,  16,  17,  and  18,  of  longitude,  along 
with  the  arguments  2,  3,  4.  5,  and  6,  heretofore  used,  take  the  cor- 
responding equations  froti.  Table  LXVII.     Find  the  sum  of  the 

37 


ASTRONOMICAL  PROBLEMS. 

whole,  omitting  the  constant  3",  and  proceed  as  in  the  rule  already 
given. 

To  obtain  the  motion  in  longitude  for  the  hour  which  precede! 
or  follows  the  given  time,  with  the  arguments  of  Tables  LXX, 
LXXII,  and  LXXIV,  take  the  equations  from  Tables  LXXVI 
and  LXX VII.  Also,  with  the  arguments  of  Evection,  Anomaly, 
Variation,  and  Reduction,  take  the  equations  from  Table  LXX VIII. 
Find  the  sum  of  all  these  equations.  Then,  for  the  hour  which  fol- 
lows the  given  time,  add  this  sum  to  the  hourly  motion  at  the  given 
time  already  found,  and  subtract  2".0 ;  for  the  hour  which  pre- 
cedes, subtract  it  from  the  same  quantity,  and  add  2".0. 

It  will  expedite  the  calculation  to  take  the  equations  of  the  sec- 
ond order  from  the  tables  at  the  same  time  with  those  of  the  first 
order  which  have  the  same  arguments. 

For  the  Hourly  Motion  in  Latitude.  The  moon's  hourly  mo- 
tion in  latitude  may  be  had  more  exactly  by  taking  with  the  argu- 
ments of  Latitude  V,  VI,  &c.  to  XII,  the  corresponding  equations 
from  Table  LXXX,  and  finding  the  sum  of  these  and  the  other 
equations  of  the  hourly  motion  in  latitude. 

To  obtain  the  moon's  motion  in  latitude  for  the  hour  which  pre- 
cedes or  follows  the  given  time,  with  the  Argument  I  of  Latitude, 
take  the  equation  from  Table  LXXXIV,  and  with  this  equation 
and  the  sum  of  all  the  equations  of  the  hourly  motion  in  longitude 
except  the  last  two,  take  the  equation  from  Table  LXXXV.  Find 
the  sum  of  these  two  equations.  Then,  for  the  hour  which  follows 
the  given  time,  add  this  sum  to  the  Hourly  Motion  in  Latitude  al- 
ready found,  taken  with  its  sign,  and  subtract  1".3;  and  for  the 
hour  which  precedes,  subtract  it  from  the  same  quantity,  and  add 
1".3. 

It  will  also  be  more  exact  to  enter  Table  LXXXI  with  the  sum 
of  all  the  equations  of  Tables  LXXIX  and  LXXX,  diminished 
by  1",  instead  of  the  equation  of  Table  LXXIX,  for  the  argument 
at  the  side.  The  numbers  over  the  tops  of  the  columns  in  Table 
LXXXI  are  the  common  differences  of  the  consecutive  numbers 
in  the  columns.  The  numbers  in  the  last  column  are  the  common 
differences  of  the  consecutive  numbers  in  the  same  horizontal  line. 

Exam.  1.  Required  the  moon's  longitude,  latitude,  equatorial 
parallax,  semi-diameter,  and  hourly  motions  in  longitude  and  lati- 
tude, on  the  14th  of  October,  1838,  at  6h.  54m.  34s.  P.  M.  mean 
time  at  New  York. 

Mean  time  at  New  York,  October,         14d-   6h<  54m-  34s- 
Diff.  of  Long.     ....  4    56      4 

Mean  time  at  Greenwich,  October,        14   11    50    38 


TO  FIND  THE  MOON  8  LONGITUDE,  ETC. 


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TO  FIND  THE  MOON  S  LONGITUDE,  ETC. 


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TO  FIND  THE  MOON?S  REDUCED  PARALLAX,  ETC.      295 

Exam.  2.  Required  the  moon's  longitude,  latitude,  equatorial 
parallax,  semi-diameter,  and  hourly  motions  in  longitude  and  lati- 
tude, on  the  9th  of  April,  1838,  at  8h.  58m.  19s.  P.  M.  mean  time 
at  Washington. 

Ans.  Long.  6s-  19°  45'  31".2;  lat.  36'  21".9  S. ;  equat.  par. 
54'  36".3  ;  semi-diameter  14'  52"  .7  ;  hor.  mot.  in  Ion*.  30'  15".2 ; 
and  hor.  mot.  in  lat.  2'  47".0,  tending  south.* 


PROBLEM  XV. 

The  Moon's  Equatorial  Parallax,  and  the  Latitude  of  a  Place, 
being  given,  to  find  the  Reduced  Parallax  and  Latitude. 

With  the  latitude  of  the  place,  take  the  reductions  from  Table 
LXIV,  and  subtract  them  from  the  Parallax  and  Latitude. 

Exam.  1.  Given  the  equatorial  parallax  55'  15",  and  the  lati- 
tude of  New  York  40°  42'  40"  N.,  to  find  the  reduced  parallax  and 
latitude. 

Equatorial  parallax,     ...         55'  15" 
Reduction,  .         .         ,         .        5 

Reduced  parallax,        .         .         .         55  10 

Latitude  of  New  York,        .  40°  42'  40;/  N. 

Reduction,          .         .         .         .          11  20 


Reduced  Lat.  of  New  York,          40    31    20 

2.  Given  the  equatorial  parallax  60'  36",  and  the  latitude  of 
Baltimore  39°  17'  23"  N.,  to  find  the  reduced  parallax  and  latitude. 

Ans.    Reduced  par.  60'  32",  and  reduced  lat.  39°  6'  9". 

3.  Given  the  equatorial  parallax  57'  22",  and  the  latitude  of 
New  Orleans  29°  57'  45"  N.,  to  find  the  reduced  parallax  and  lat- 
itude. 

Ans.  Reduced  par.  57'  19",  and  reduced  lat.  29°  47'  50". 


PROBLEM  XVI. 

To  find  the  Longitude  and  Altitude  of  the  Nonagesimal  Degree 
of  the  Ecliptic,  for  a  given  time  and  place. 

For  the  given  time,  reduced  to  mean  time  at  Greenwich,  find  the 
sun's  mean  longitude  and  the  argument  N  from  Tables  XVIII, 
XIX,  XX,  and  XXI.  To  the  sun's  mean  longitude,  apply  accord- 
ing to  its  sign  the  nutation  in  right  ascension,  taken  from  Table 

*  The  smaller  equations  were  omitted  in  working  this  example. 


296  ASTRONOMICAL  PROBLEMS. 

XXVII  with  argument  N ;  and  the  result  will  be  the  right  ascen- 
sion of  the  mean  sun,  (see  Art.  45,)  reckoned  from  the  true  equi- 
nox. 

Reduce  the  mean  time  of  day  at  the  given  place,  expressed  as 
tronomically,  to  degrees,  &c.,  and  add  it  to  the  right  ascension  of 
the  mean  sun  from  the  true  equinox.     The  sum,  rejecting  360°, 
when  it  exceeds  that  quantity,  will  be  the  right  ascension  of  the 
midheaven,  or  the  sidereal  time  in  degrees. 

Next,  find  the  reduced  latitude  of  the  place  by  Problem  XV ; 
and  when  it  is  north,  subtract  it  from  90°  ;  but  when  it  is  south, 
add  it  to  90°.  The  sum  or  difference  will  be  the  reduced  distance 
of  the  place  from  the  north  pole. 

Also,  take  the  obliquity  of  the  ecliptic  for  the  given  year  from 
Table  XXII.* 

These  three  quantities  having  been  found,  the  longitude  and  alti- 
tude of  the  nonagesimal  degree  may  be  computed  from  the  follow- 
ing formulae : 

log.  cos  J  (H  -  «)  —  log.  cos  i  (H  +  ")  =  A  .  .  .  (1); 
log.  tang  J  (H  —  w)  +  10  -  log.  tang  |  (H  +  «)  =  B  .  .  .  (2) ; 
log.  tang  E  =  A  +  log.  tang  £  (S  —  90°)  •  •  •  (3)  J 
log.  tang  F  =  log.  tang  E  +  B  .  .  .  (4)  ; 
N  =  E+F  +  90°  .  .  .  (5); 

log.  tang  ±h  =  log.  cos  E  +  log-  tang  4-  (H  -f  w)  +  ar.  co.  log. 

cos  F  -  20  ...  (6). 
in  which 

H  =  the  reduced  distance  of  the  place  from  the  north  pole ; 

w  =  the  Obliquity  of  the  Ecliptic  ; 

S  =  the  Sidereal  Time  converted  into  degrees ; 

N  =  the  required  Longitude  of  the  Nonagesimal ; 

h  =  the  required  Altitude  of  the  Nonagesimal ; 

E  and  F  are  auxiliary  angles. 

We  first  find  the  logarithmic  sums  A  and  B.  With  these  we  de- 
termine the  angles  E  and  F  by  formulae  (3)  and  (4),  and  with  these 
again  N  and  h  by  formulae  (5)  and  (6). 

The  angles  E,  F,  are  to  be  taken  Jess  than  180°;  and  less  or 
greater  than  90°,  according  as  the  sign  of  their  tangent  proves  to  be 
positive  or  negative. 

Note  1 .  In  case  the  given  place  lies  within  the  arctic  circle,  we 
must  take,  in  place  of  formula  (5),  the  following : 
N  =  E  -  F  +  90°. 

*  If  great  precision  is  required,  the  apparent  obliquity  is  to  be  used  in  place  of  the 
mean.  (See  Prob.  X.) 


TO  FIND  THE  LONG.  AND  ALT.  OF  NONAGESIMAL  DEGREE.     297 

Note  2.  As  the  obliquity  of  the  ecliptic  varies  but  slowly  from 
year  to  year,  the  values  which  have  once  been  found  for  the  loga- 
rithms A,  B,  and  log.  tang  i  (H  +  w)  (C),  will  answer  for  several 
years  from  the  date  of  their  determination,  unless  very  great  accu- 
racy is  required. 

Note  3.  The  angle  h  derived  from  formula  (6),  is  the  dis- 
tance of  the  zenith  of  the  given  place  from  the  north  pole  of  the 
ecliptic.  This  is  not  always  equal  to  the  altitude  of  the  nonagesi- 
mal.  Throughout  the  southern  hemisphere,  and  frequently  in  the 
northern  near  the  equator,  it  is  the  supplement  of  the  altitude.  In 
employing  this  angle  in  the  following  Problem,  it  is,  however,  for  the 
sake  of  simplicity,  called  the  altitude  of  the  nonagesimal  in  all  cases. 

Exam.  1 .  Required  the  longitude  and  altitude  of  the  nonagesi- 
mal degree  of  the  ecliptic  at  New  York,  on  the  1 8th  of  September, 
1838,  at  3h.  52m.  56s.  P.  M.  mean  time. 

The  sun's  mean  longitude  taken  from  the  tables,  for  the  given 
time,  is  5s-  27°  19'  17",  and  the  argument  N  is  987.  The  nutation 
taken  from  Table  XXVII  with  argument  N  is  —  I".  Hence,  the 
right  ascension  of  the  mean  sun,  reckoned  from  the  true  equinox, 
is  5s-  27°  19'  16".  The  given  time  of  day,  expressed  astronomi- 
cally, is  3h.  52m.  56sec. ;  which  in  degrees  is  58°  14'  0". 

The  reduced  latitude  of  New  York,  found  by  Problem  XV,  is 
40°  31'  20",  and  this  taken  from  90°  leaves  the  polar  distance  49° 
28'  40".  The  obliquity  of  the  ecliptic,  derived  from  Table  XXII, 
is  23°  27'  37". 

Given  time  in  degrees,     .         .         .     58°  14'    0" 
R.  Asc.  of  mean  sun,        .         .         .  177    19  16 

Sidereal  time  in  degrees  (S),     .         .  235    33  16 

90 


2)145  33  16 
H    .  .  49°  28' 40" 
u    .  .  23  27  37         i  (S  —  90)  72  46  38 

Diff  .  .  26  1  3 
Sum    .  72  56  17 


Jdiff.  .     .13     031     .      cos.  9.98870     .     tan.  +  10,19.36366 
sum.     .    36  28     8  cos.  9.90535     .     tan.       C.  9.86871 


A.  0.08335  B.  9.49495 

i(S  -  90°)  72  46  38  .   tan.  0.50866 

E    .    75  38  55  .   tan.  0.59201  .'  cos.     9.39422 

B.  9.49495  C.  9.86871 

F    .    50  41  55  .   tan  0.08696  .  ar.  co.  cos.  0.19832 
90  0  0 

alt.  non.  16°  7'  54"  .     tan.  9.46126 


long.  non.  216  20  50 


alt.  non.  32  15  48 

38 


298  ASTRONOMICAL  PROBLEMS. 

2.  Required  the  longitude  and  altitude  of  the  nonagesimal  de- 
gree of  the  ecliptic  at  New  York,  on  the  10th  of  May,  1838,  at 
llh.  33m.  56sec.  P.  M.  mean  time. 

Ans.  Long.  200°  12'  23",  and  alt.  37°  0'  34". 

PROBLEM  XVII. 

To  find  the  Apparent  Longitude  and  Latitude,  as  affected  by 
Parallax,  and  the  Augmented  Semi-diameter  of  the  Moon ;  the 
Moon's  True  Longitude,  Latitude,  Horizontal  Semi-diameter, 
and  Equatorial  Parallax,  and  the  Longitude  and  Altitude  of 
the  Nonagesimal  Degree  of  the  Ecliptic,  being  given. 

We  have  for  the  resolution  of  this  Problem  the  following  for- 
mulae : 

log.  x  =  log.P-f  log.  cos^+ar.  co.log.cos  X—  10  ...  (1); 

c  =  log.  x  +  log.  tang  h  —  10  ...  (2) ; 
log.  u  =  c  +  log.  sin  K  —  10  ...  (3) ; 
log.w'  =  c+log.  sin(K  +  w)-  10  .  .  .  (4); 
log.jp  =  c+log.  sin (K  +  O-  10  .  .  .  (5); 
Appar.  long.  =  true  long.  +p  .  .  .  (6) ; 

log.  tang  X'  =  log.  p  +  ar.  co.  log.  cos  X  +  ar.  co.  log.  u  +  log. 

sin(X-#)-  10*  ...  (7); 

log.  v  =  log.  P  +  log.  cos  h  +  log.  cos  X7  —  10   ...  (8) ; 
log.  z  —  log.  v  -f  log.  tang  h  +  log.  tang  X7  -f  log.  cos 

(K  +  fc>)-30  .  .  .  (9); 
if  =v  —  z  .  .  .  (10); 
Appar.  lat.     =  true  lat.  —  «•  .  .  .  (11); 

log.R'  =  leg.  p  +  ar.  co.  log.  cos  X  +  ar.  co.  log.  u  +  log. 

cos  X7  -flog.  R  —  10  ...  (12) : 
in  which 

P  =  the  Reduced  Parallax  of  the  Moon ; 
h  =  the  Altitude  of  the  Nonagesimal ; 
X  =  the  True  Latitude  of  the  Moon  (minus  when  south) ; 
K  =  the  Longitude  of  the  Moon,  minus  the  longitude  of  the  No- 
nagesimal ; 

p  =  the  required  Parallax  in  Longitude  ; 
X'  =  the  approximate  Apparent  Latitude  of  the  Moon ; 
if  =  the  required  Parallax  in  Latitude  ; 
R  =  the  True  Semi-diameter  of  the  Moon ; 
R;  =  the  Augmented  Semi-diameter  of  the  Moon ; 
x,  u,  u',  v,  z,  are  auxiliary  arcs. 

*  Formula  (7)  will  be  rendered  more  accurate  by  adding  to  it  the  ar.  co.  cos 
*  — 10,  and  will  generally  give  the  apparent  latitude  with  sufficient  accuracy; 
thus  rendering  formulae  (8),  (9),  (10),  and  (11)  unnecessary. 


TO  FIND  THE  MOON?S  APPARENT  LONG.  AND  LAT.  299 

Formulae  (1),  (2),  (3),  (4),  and  (5),  being  resolved  in  succession, 
we  derive  the  apparent  longitude  from  formula  (6) ;  then  the  appa- 
rent latitude  from  equations  (7),  (8),  (9),  (10),  (11);  and  lastly, 
the  augmented  semi-diameter  from  equation  (12.) 

The  latitude  of  the  moon  must  be  affected  with  the  negative 
sign  when  south ;  and  the  apparent  latitude  will  be  south  when  it 
comes  out  negative.  In  performing  the  operations,  it  is  to  be  re- 
membered that  the  cosine  of  a  negative  arc  has  the  same  sign  as 
the  cosine  of  a  positive  arc  of  an  equal  number  of  degrees  ;  but 
that  the  sine  or  tangent  of  a  negative  arc  has  the  opposite  sign  from 
the  sine  or  tangent  of  an  equal  positive  arc.  Attention  must  also 
be  paid  to  the  signs  in  the  addition  and  subtraction  of  arcs.  Thus, 
two  arcs  affected  with  essential  signs,  which  are  to  be  added  to 
each  other,  are  to  be  added  arithmetically  when  they  have  like 
signs,  but  subtracted  if  they  have  unlike  signs  ;  and  when  one  arc 
is  to  be  taken  from  another,  its  sign  is  to  be  changed,  and  the  two 
united  according  to  their  signs.  An  arithmetical  sum,  when  taken, 
will  have  the  same  sign  as  each  of  the  arcs ;  and  an  arithmetical 
difference  the  same  sign  as  the  greater  arc. 

The  use  of  negative  arcs  may  be  avoided,  though  the  calculation 
would  be  somewhat  longer,  by  using  the  true  polar  distance  d,  and 
the  approximate  apparent  polar  distance  d',  in  place  of  X  and  X', 
substituting  sin  d  for  cos  X,  cos  (d  •}-  x)  for  sin  (X  —  oc\  sin  d'  for 
cos  X',  log.  co-tang  d'  for  log.  tang  X' ;  and  observing  that  p  is 
to  be  subtracted  from  the  true  longitude  in  case  the  longitude  of 
the  nonagesimal  exceeds  the  longitude  of  the  moon  ;  that  z,  when 
it  comes  out  negative,  is  to  be  added  to  v,  which  is  always  positive 
to  the  north  of  the  tropic,  otherwise  subtracted  ;  and  that  the  par- 
allax in  latitude  is  to  be  applied  according  to  its  sign  to  the  true 
polar  distance. 

In  seeking  for  the  logarithms  of  the  trigonometrical  lines,  it  will 
be  sufficient  to  take  those  answering  to  the  nearest  tens  of  seconds. 

Note  1 .  When  great  accuracy  is  not  desired,  u'  may  be  taken 
for  p,  from  which  it  can  never  differ  more  than  a  fraction  of  a 
second. 

Note  2.  In  solar  eclipses  the  moon's  latitude  is  very  small,  and 
formula  (7)  may  be  changed  into  the  following : 

log.  X'  =  log.  p +ar.  co.  log.  cos  X+ar.  co.  log.  u  -Hog.  (X  —  a?)— 10 

and  cos  X'  omitted  in  formula  (12)  without  material  error. 

Formulae  (8),  (9),  (10),  and  (11),  may  also  now  be  dispensed 
with,  unless  very  great  precision  is  desired,  and  the  value  of  X' 
given  by  the  above  formula  taken  for  the  apparent  latitude. 

It  is  to  be  observed  also,  that  in  eclipses  of  the  sun  P  is  taken 
equal  to  the  reduced  parallax  of  the  moon  minus  the  sun's  horizon- 
tal parallax.  By  this  the  parallax  of  the  sun  in  longitude  and  lati- 
tude is  referred  to  the  moon,  and  the  relative  apparent  places  of 
the  sun  and  moon  are  correctly  obtained,  without  the  necessity  of 


300  ASTRONOMICAL  PROBLEMS. 

a  separate   computation  of  the  sun's  parallax  in  longitude   and 
latitude. 

Exam.  1 .  About  the  time  of  the  middle  of  the  occultation  of  the 
star  Antares,  on  the  10th  of  May,  1838,  the  moon's  longitude,  by 
the  Connaissance  des  Terns,  was  247°  37'  6".7;  latitude  4°  14' 
14".7  S. ;  semi-diameter  15'  24".2 ;  and  equatorial  parallax  56' 
31  ".7 ;  and  the  longitude  of  the  nonage simal  at  New  York  was 
200°  12'  23"  ;  the  altitude  37°  0'  34"  ;  required  the  apparent  Ion- 
gitude  and  latitude,  and  the  augmented  semi-diameter  of  the  moon 
at  New  York,  at  the  time  in  question. 

Equat.  par.     56' 31".7  Moon's  long.     247°  37'    7" 

Reduction  4  .6  Long,  nonag.    200  12  23 

P  =  56  27  .1  K  =  47  24  44 

h  =  37     0  34 
X  =  -4  14  14.7 

P  3387".!     .    .  log.  3.52983 

h    .    .    .37°  0'34"     .    .  cos.  9.90230 

a.  3.43213 
X    .    .     —41415       ar.  co.  cos.  0.00119 

x  45  12  .  2712"   .  log.  3.43332 

h    .    .    .  37  0  34  .    .    .  tan.  9.87725 

c.   3.31057 
K    .    .    .  47  24  44  .    .      sin.  9.86701 

tt    ;    .    .     25  5  .  1505"   .  log.  3.17758 

c.   3.31057 
47  49  49  sin.  9.86991 


u'  25  15  .  1515".2  .  log.  3.18048 

c.      3.31057 
K  +  u'          .         .     47    49  59     .         .         .     sin.  9.86993 

p  25  15.3  .  1515".3  .     log.  3.18050 

True  long.   .         .  247    37     6.7 

Appar.  long.         .  248      2  22.0 

p log.  3.18050 

x— x  .    .    .  — 4  59  27  .    .    .  sin.  8.93957- 
X    .    .   /%>   .    .    .    .  ar.  co.  cos.  0.00119 
tt  .       .  ^.    .  ar.  co.  log.  6.82242 

X'  ,—51  10  tan.  8.94368- 


301 


5°     1'  10"  .         .         .      cos.  9.99833 

a.     3.43213 


V  44  54.4  .  2694".4  .  log.  3.43046 

h         .........  tan.  9.87725 

X'         ........  tan.  8.94368— 

K  +  |  .     47    37  22     .         .         .  cos.  9.82867 


z         .         .         .          —  2     0.2   .    120".2  .     log.  2.08006  - 
v-z    .         .         .  46  54.6 

v—z  (sign  changed)      —46  54.6 
Truelat.  .    —4  14  14.7 


Appar.  lat.  ..51     9.3  S. 

p log.  3.18050 

X ar.  co.  cos.  0.00119 

u ar.  co.  log.  6.82242 

X7 cos.  9.99833 

R  15  24.2  .     924".2  .      log.  2.96577 

Augm.  semi-diam.  15  29.4  .     929".4  .     log.  2.96821 

Exam.  2.  About  the  middle  of  the  eclipse  of  the  sun  on  the  18th 
of  September,  1838,  the  moon's  longitude  was  175°  29'  19".0, 
latitude  47'  47".5,  equatorial  parallax  53'  53".5,  and  semi-diame- 
ter 14'  41  ".1  ;  and  the  longitude  of  the  nonagesimal  at  New  York 
was  216°  20'  50",  the  altitude  32°  15'  48":  required  the  apparent 
longitude  and  latitude,  and  the  augmented  semi-diameter  of  the 
moon. 

Equat.  paral.     53'  53".5  Moon's  long.      175°  29'  19'- 

Reduction,  4  .4          .       Long,  nonag.    216  20  50 


Sun's 

53  49  .1 

paral.      8  .6 

K  =  -40  51  31 
h  =  32  15  48 

X  =   0  47  47.5 

P  =  53  40  .5 

P 

h 
X 

.  3220' 
32°  15'  48"  . 
47  47.5  . 

'.5  .    .    log.  3.50792 
cos.  9.92716 
ar.  co.  cos.  0.00004 

X 

h 

45  23.5.  . 
32  15  48   . 

2723".5  .    log.  3.43512 
tan.  9.80023 

K 

.  -40  51  31   . 

c.  3.23535 

sin.  9.81570- 

—1845       .1126"      .         log.  3.05 L05- 


302 


ASTRONOMICAL    PROBLEMS. 


True  lon. 


-41°  10'  16"     .         . 

—18  52.9    .  1132;/.9  . 


c. 

sin. 


3.23535 
9.81844- 


log.  3.05379- 


—41   10  24 

—  18  52.9 

175  29  19.0 

175  10  26.1 


c. 

sin. 


3.23535 

9.81844- 


long. 

Appar.  long. 

P 

X  ...... 

u  ...... 

X-a?    .         .         2/24//.0    .     144".0 

Appar.  latitude      2'  24".9  N.    144".9 


P 
X 

u 
R 


1132".9  .         log.  3.05379— 


ar.  co. 
ar.  co, 


ar.  co, 
ar.  co. 


Augm.  semi-diam.  14  46  .7   .     886".7  . 


log. 
cos. 
log. 
log. 


3.05379 
0.00004 
6.94895 
2.15836 


log.  2.16114 


log. 
cos. 
log. 
log. 


3.05379 
0.00004 
6.94895 
2.94502 


log.  2.94780 


PROBLEM   XVIII. 

To  find  the  Mean  Right  Ascension  and  Declination,  or  Longitude 
and  Latitude  of  a  Star,  for  a  given  time,  from  the  Tables. 

Take  the  difference  between  the  given  year  and  1 840.  Then 
seek  in  Table  XV  for  the  fraction  of  the  year  answering  to  the 
given  month  and  days,  and  add  it  to  this  difference,  if  the  given 
time  is  after  the  beginning  of  the  year  1840;  but  if  it  is  before, 
subcract  it.  Multiply  the  sum  or  difference  by  the  annual  variation 
given  in  the  catalogue,  (Table  XC,  or  XCII,)  and  the  product  will 
be  the  variation  in  the  interval  between  the  given  time  and  the 
epoch  of  the  catalogue.  Apply  this  product  to  the  quantity  given 
in  the  catalogue,  according  to  its  sign,  if  the  given  time  is  after 
the  beginning  of  the  year  1840,  but  with  the  opposite  sign  if  it  is 
before,  and  the  result  will  be  the  quantity  sought. 

Exam.  1.  Required  the  mean  right  ascension  and  declination  of 
the  star  Sirius  on  the  15th  of  August,  1842. 

Interval  between  given  time  and  beginn.  of  1840,  (t,)       2.619yrs, 
Annual  variation  of  right  ascension;      »§£_•      •         •  2.646s 

Variation  of  right  ascension  for  interval  f,         .        .          6.93s. 


TO  FIND  A  STAR'S  ABERR.  IN  RIGHT  ASCENSION,  ETC. 

A  similar  operation  gives  for  the  variation  of  declination  in  the 
same  interval,  11". 65. 

Mean  right  ascen., beginning  of  1840,  Table  XC,     6h-  38™-  5.76s- 
Variation  for  interval  t,     .         .         .         .         .  +6.93 


Mean  right  ascension  required,          .         .  6  38  12.69 

Mean  declination,beginning  of  1840,          .         .  16°  30'    4".79S, 
Variation  for  interval  t, + 1 1   .65 


Mean  declination  required,       .         ,         .         .  16  30  16  .44  S. 

2.  Required  the  mean  longitude  and  latitude  of  Aldebaran  on 
the  20th  of  October,  1838. 

Interval  between  given  time  and  begin,  of  1840,  (t)          1.200yrs. 
Annual  variation  of  longitude,         ....         50".210 

Variation  of  longitude  for  interval  t,  60". 2 

A  similar  operation  gives  for  the  variation  of  latitude  in  the  same 
interval  0".4. 

Mean  longitude, beginning  of  1840,          .         2s-   7°    33'    5".9 
Variation  for  interval  t,  ...  —  1     0  .2 


Mean  longitude  required,        .         .         .         2     7     32     5  .7 

Mean  latitude, beginning  of  1840,    .         .  5°    28'  38".0  S. 

Variation  for  interval  t,  .  +  0  .4 


Mean  latitude  required,  .         .         .         .  5     28  38  .4  S. 

3.  Required  the  mean  right  ascension  and  declination  of  Capella 
on  the  9th  of  February,  1839  ? 

Ans.  Mean  right  ascension  5h-  4ra-  48.74"-,  and  mean  declination 
45°  49'  38".53  N. 

4.  Required  the  mean  longitude  and  latitude  of  Aldebaran  on 
the  1 6th  of  April,  1845? 

Ans.  Mean  longitude  2s-  7°  37'  31".4,  and  mean  latitude  5°  28' 
36".2. 


PROBLEM  XIX. 

To  find  the.  Aberrations  of  a  Star  in  Right  Ascension  and  Decli 
nation,  for  a  given  Day. 


This  problem  may  be  resolved  for  any  of  the  stars  in  the 
logue  of  Table  XC  by  means  of  the  following  formulae  : 


304  ASTRONOMICAL  PROBLEMS. 

log.  (aber.  in  right  ascen.)  ==  M  +  log.  sin  (O  +  9j  —  10. 
log.  (aber.  in  declin.)          =  N  +  log.  sin  (O  +  6)  —  10, 

in  which  M,  N,  are  constant  logarithms,  O  the  longitude  of  the  sun 
on  the  given  day,  and  9,  0,  auxiliary  angles.  M,  N,  and  the  an- 
gles 9,  69  are  given  for  each  of  the  stars  in  the  catalogue,  in 
Table  XCI.  O  may  be  derived  from  an  ephemeris  of  the  sun, 
or  it  may  be  computed  from  the  solar  tables  by  Problem  IX. 

Exam.  1.  What  was  the  amount  of  aberration,  in  right  ascen- 
sion and  declination,  of  a  Orionis  on  the  20th  of  December,  1 837, 
the  sun's  longitude  on  that  day  being  8s-  28°  28'  ? 

Right  Ascension. 
Table  XCI,  9       .         6s-    3°  13'     M  .        .         0.1361 

O     .         8    28    28 


O+9.3      1    41          .         .sin.  9.9998 


Aberration  =  1".37     .         .         .         .log.  0.1359 

Declination. 
Table  XCI,  6        .         88-  28°  23'     N.        .         0.7521 

O  8   28   28 


O-H   .     5   26   51          .         .sin.  8.7399 


Aberration  =  0".31      ....  log.  1.4920 

2.  Required  the  aberrations  in  right  ascension  and  declination 
of  a  Andromedae  on*the  1st  of  May,  1838,  the  sun's  longitude  be- 
ing I8-  10°  38'. 

Ans.  Aberr.  in  right  ascension  —  1".07,  and  aberr.  in  declina- 
tion -  11".69. 


PROBLEM  XX. 

To  find  the  Nutations  of  a  Star  in  Right  Ascension  and  Declines 
tion,  for  a  given  Day. 

This  Problem  may  be  solved  by  means  of  the  formulae, 
log.  (nuta.  in  right  asc.)  =  M'  +log.  sin  (ft  4- 9')  —  10 ; 
log.  (nuta.  in  declin.)       =  N'  -flog,  sin  (ft  +6')  —  10; 

in  which  M',  N',  are  constant  logarithms,  ft  the  mean  longitude  of 
the  moon's  ascending  node,  and  9',  &',  auxiliary  angles.  M',  N', 
and  the  angles  9',  £',  are  given  for  each  of  the  stars  in  the  cata- 
logue, in  Table  XCI.  The  mean  longitude  of  the  moon's  ascend- 
ing node  is  given  for  every  tenth  day  of  the  year  in  the  Nautical 
Almanac,  page  266,  and  may  be  easily  found  for  any  intermediate 


TO  FIND  A  STAR'S  NOTATION  IN  RIGHT  ASCEN.,  ETC.       305 

day  from  the  daily  motion  inserted  at  the  foot  of  the  column  of 
longitudes.  It  may  also  be  had  by  finding  the  supplement  of  the 
moon's  node,  for  the  given  time,  from  the  lunar  tables,  and  sub- 
tracting it  from  12s-  0°  7'. 

Exam.  1.  What  was  the  amount  of  the  nutation,  in  right  ascen- 
sion and  declination,  of  a  Orionis  on  the  20th  of  December,  1 837, 
the  mean  longitude  of  the  moon's  node  on  that  day  being  18°  54'  ? 

Right  Ascension. 

TableXCI,  9'         .     6s-   0°    15'     M'     .         .     0.0481 
ft         .     0  18     54 


+9'.     6  19       9      .         .       sin.  9.5159 


Nutation  =  -  0".37        .       log.  1.5640- 

Declination. 

Table  XCI,  *'          .     3s-   2°    37'     N'     .         .     0.9657 
ft         .     0  18     54 


&+*'.    S  21     31      .         .       sin.  9.9686 


Nutation  =       8".60        .       log.  0.9343 

2.  Required  the  nutations  in  right  ascension  and  declination  of 
a  Andromedae  on  the  1st  of  May,  1838. 

Ans.  Nutation  in  right  ascension  —  0".54,  and  nutation  in  de- 
cimation —  I"  A3. 

Note.  When  the  apparent  place  of  a  star  is  desired  with  great 
accuracy,  the  solar  nutations  must  also  be  estimated  and  allowed 
for.  These  may  be  determined  by  repeating  the  process  for  find- 
ing the  lunar  nutations,  only  using  twice  the  sun's  longitude  in 
place  of  the  longitude  of  the  moon's  node,  and  multiplying  the  re- 
sults by  the  decimal  .075. 

The  calculation  of  the  solar  nutations  in  Example  1st,  is  as  fol- 
lows : 

Right  Ascension. 

Table  XCI,  9'    -        -    6s-  0°    15'    M'    .        .    0.0481 
2O     .  5  26     56 


2O  +  9'.  11  27     11   f-^       .      sin.  8.6914— 

—  0/;.05    .     log.~2".7395- 
.075 

Solar  Nutat.  =  -  0".00 
39 


306  ASTRONOMICAL  PROBLEMS* 

Declination. 

Table  XCI,d'    .         .     3s-   2°    37'     N'     .         .     0.9657 
20  5  26     56 


2O+4'.     8  29     33      .         .     sin.  10.0000— 

—  9".24    .    *^     0.9657— 
.075 

Solar  Nutat.  =  -  0".69 

In  Example  2d,  we  find  for  the  solar  nutation  in  right  ascension, 
—  0".08,  and  for  the  solar  nutation  in  declination,  —  0".51. 


PROBLEM  XXI. 

To  find  the  Apparent  Right  Ascension  and  Declination  of  a  Star, 
on  a  given  Day. 

Find  the  mean  right  ascension  and  decimation  for  the  given  day 
by  Problem  XVIII ;  then  compute  the  aberrations  in  right  ascen- 
sion and  declination  by  Problem  XIX,  and  the  lunar  and  solar  nu- 
tations in  right  ascension  and  declination  by  Problem  XX.  Apply 
the  aberrations  and  nutations  according  to  their  signs,  to  the  mean 
right  ascension  and  declination  on  the  given  day,  observing  that  the 
declination  when  south  is  to  be  marked  negative,  and  the  results 
will  be  the  apparent  right  ascension  and  declination  sought. 

Exam.  1 .  What  was  the  apparent  right  ascension  and  declina- 
tion of  a  Orionis  on  the  20th  of  December,  1837  ? 

h.  m.      s.  °     '       " 

Table  XC,  M.  right  ascen.    5  46  30.71     M.  dec.  7  22  17.14N. 
Variations  -  6.59  —2.42 


5  46  24.12  7  22  14.72 

Aberr.   .     [  iv  *         +1.37  wU4^       +0.31 

Lun.  nutat.    .             —0.37  .         .           +8.60 

Sol.  nutat.                       0.00  .           -  0.69 


App.  right  asc.  5  46  25.12    App.dec.7  22  22.94N. 

2.  Required  the  apparent  right  ascension  and  declination  of 
a  Andromedae  on  the  1st  of  May,  1838. 

Ans.  Appar.  right  ascen.  Oh.  Om.  0.90s.,  and  appar.  dec. 
28°  11'  39".92. 


TO  FIND  A  STAR'S  ABERRATION  IN  LONGITUDE,  ETC.       307 


PROBLEM  XXII. 

To  find  the  Aberrations  of  a  Star  in  Longitude  and  Latitude,  for 
a  given  Day, 

The  formulae  for  the  computation  are, 

log.  (aber.  in  long.)  =  1.30880  +  log.  cos  (6s.  +  O  —  L)  +  ar. 

co.  log.  cos  X  —  10  ; 

log.  (aber.  in  lat.)  =  1.30880  -f  log.  sin  (6s.  +  O  -  L)  +  log. 

sin  X  —  20 ; 

in  which  O  =  longitude  of  the  sun  on  the  given  day  ;  L  =  mean 
longitude  of  the  star ;  and  X  =  mean  latitude  of  the  star. 

Exam.  1 .  Required  the  aberrations  in  longitude  and  latitude  of 
Antares  on  the  26th  of  February,  1838,  the  sun's  longitude  on  that 
day  being  11s-  7°  29'. 

By  Prob.  XVIII,  L  =  8s-  7°  30',     and  X  =          4°  32'  S. 
6s. +  O     .     17     7   29       Const,  log.    1.3088 

6s.  +  O  -  L   8  29   59    .       '  -.'     cos.   6.4637  — 
X      .         .  4  32   .    ar.  co.  cos.   0.0014 

Aberr.  in  long.  =  -0".00  .  log.  1.7739  — 

Const,  log.    1.3088 

6s.  +  O  -  L  8*  29°  59'  .         .      sin.  10.0000  — 
X      ...          4  32    .         .      sin.    8.8978 

Aberr.  in  lat.  =  -  1;/.61  .  log,    0.2066  - 

2.  Required  the  aberrations  in  longitude  and  latitude  of  Arc- 
turus  on  the  5th  of  October,  1838,  the  sun's  longitude  being 
6"-  11°  47'. 

Ans.  Aberr.  in  long.  —  23".34,  and  aberr.  in  lat.  1".85. 
Note.  The  nutation  in  longitude  of  a  fixed  star  may  be  found 
after  the  same  manner  as  the  nutation  in  longitude  of  the  sun. 
See  Problem  IX.) 


PROBLEM  XXIII. 

To  find  the  Apparent  Longitude  and  Latitude  of  a  Star,  for  a 

given  Day. 

Find  the  mean  longitude  and  latitude  on  the  given  day  by  Prob- 
lem XVIIL  Find  also  the  aberrations  in  longitude  and  latitude  by 
Problem  XXII,  and  the  nutation  in  longitude,  as  in  Problem  lA, 
Apply  the  aberration  and  nutation  in  longitude,  according  to  their 


308  ASTRONOMICAL  PROBLEMS. 

signs,  to  the  mean  longitude,  and  the  result  will  be  the  apparent 
longitude ;  and  apply  the  aberration  in  latitude  according  to  its 
sign,  to  the  mean  latitude,  and  the  result  will  be  the  apparent 
latitude. 

Exam.  1.  Required  the  apparent  longitude  and  latitude  of  An. 
tares  on  the  26th  of  February,  1838. 

Table  XC,  M.  long.   8a'  7°  31'  45".2         M.  lat.  4°  32'  51".6  S. 
Var.  —  1  32  .57  0  .78 


8     7  30  12  .63         .       .4  32  50  .82 
Aberr.     .  0  .00         .  —  1  .61 

Nutat.  -  4  .40 


App.long.  8     7  30     8  .23    App.  lat.  4  32  49  .21  S. 

2.  Required  the  apparent  longitude  and  latitude  of  Arcturus  on 
the  5th  of  October,  1838. 

Ans.  Appar.long.  6s-  21°  58'  37" .4,  andappar.  lat. 30°  51'  19".  1. 

PROBLEM   XXIV. 

To  compute  the  Longitude  and  Latitude  of  a  Heavenly  Body  from 
its  Right  Ascension  and  Declination,  the  Obliquity  of  the  Eclip- 
tic being  given. 

This  Problem  may  be  solved  by  means  of  the  following  for- 
mulae : 

log.  tang  x  —  log.  tang  D  +  ar.  co.  log.  sin  R ; 

log.  tang L=log.  cos  (x— w)  +  log.  tang  R  +  ar.  co. log.  cos  x— 10; 

log.  tang  X  =  log.  tang  (x  —  w)  +  log.  sin  L  —  10  ; 

in  which 

R  =  the  Right  Ascension  ; 

D  =  the  Declination  (minus  when  South)  ; 

L  =  the  Longitude  ; 

X  =  the  Latitude  ; 

eo  =  the  Obliquity  of  the  ecliptic ; 

x  is  an  auxiliary  arc.  It  must  be  taken  according  to  the  sign  of 
its  tangent,  but  always  less  than  180°.  The  longitude  will  always 
be  in  the  same  quadrant  as  the  right  ascension.  The  latitude  must 
be  taken  less  than  90°,  and  will  be  north  or  south,  according  as  the 
sign  is  positive  or  negative. 

Note.  When  the  mean  longitude  and  latitude  are  to  be  derived 
from  the  mean  right  ascension  and  decimation,  the  mean  obliquity 
of  the  ecliptic  is  taken.  When  the  apparent  longitude  and  latitude 
are  to  be  derived  from  the  apparent  right  ascension  and  declina- 
tion, found  as  in  Problem  XAl,  the  apparent  obliquity  is  taken. 


TO  COMPUTE  THE  RIGHT  ASCEN.  AND  DEC.  OF  A  BODY.        309 

The  mean  obliquity  of  the  ecliptic  at  any  assumed  time  is  easily 
deduced  from  Table  XXII.  The  apparent  obliquity  is  found  by 
Problem  X. 

Exam.  1.  On  the  20th  of  June,  1838,  the  right  ascension  of 
Capella  was  76°  11'  29",  the  declination  45°  49'  35"  N.,  and  the 
obliquity  of  the  ecliptic  23°  27'  37"  ;  required  the  longitude  and 
latitude. 

D  =  45°  49'  35"     .         ,         .         tan.  0.0125295 
R  =  76   11  29  ar.  co.  sin.  0.0127367 


x  =  46  39  56       .         .         .         tan.  0.0252662 

w  =  23   27  37  


x  —  u  =  23  12  19  .  .  .  cos.  9.9633623 
R=  76  11  29  .  .  .  tan.  0.6094483 
x  =  46  39  56  ar.  co.  cos.  0.1635240 


Long.  =  79   36     4  .         .         tan.  0.7363346 

L  =  79   36     4       .         .         .         sin.  9.9928075 
x  —  u  =  23   12  19       .  tan.  9.6321632 


Lat.  =  22  51  49       .         .         .         tan.  9.6249707 

2.  Given  the  right  ascension  of  Spica  199°  IT  35",  and  decli- 
nation 10°  19'  24"  S.,  and  the  obliquity  of  the  ecliptic  23°  27'  36", 
on  the  1st  of  January,  1840,  to  find  the  longitude  and  latitude. 
Ans.  Long.  201°  36'  32",  and  lat.  2°  2'  30"  S. 

PROBLEM  XXV. 

To  compute  the  Right  Ascension  and  Declination  of  a  Heavenly 
Body  from  its  Longitude  and  Latitude,  the  Obliquity  of  the 
Ecliptic  being  given. 

The  formulae  for  the  solution  of  this  problem  are, 
log.  tang  y  =  log.  tang  X  -f  ar.  co.  log.  sin  L  ; 
log. tang  R  =log.cos(y-r-w)  +  log.  tang  L  +  ar.  co.  log.  cos  y— 10; 
log.  tang  D  =  log.  tang  (y  +  w)  +  log.  sin  R  —  10 ; 
in  which 

L  =  the  Longitude  ; 

X  =  the  Latitude  (minus  when  South)  ; 

R  =  the  Right  Ascension  ; 

D  —  the  Declination  ; 

w  =  the  Obliquity  of  the  ecliptic  ; 

y  is  an  auxiliary  arc.     It  must  be  taken  according  to  the  sign  of 
its  tangent,  but  always  less  than  180°.     The  right  ascension  will 


310  ASTRONOMICAL  PROBLEMS. 

always  be  in  the  same  quadrant  with  the  longitude.  The  declina- 
tion must  be  taken  less  than  90°,  and  will  be  north  or  south,  ac- 
cording as  the  sign  is  positive  or  negative. 

Note.  The  mean  or  apparent  obliquity  of  the  ecliptic  is  taken, 
according  as  the  given  and  required  elements  are  mean  or  apparent. 

Exam.  1.  On  the  1st  of  January,  1830,  the  longitude  of  Sirius 
was  3s-  11°  44'  18",  the  latitude  39°  34'  1"  S.,  and  the  obliquity 
of  the  ecliptic  23°  27'  41 "  :  required  the  right  ascension  and  de- 
clination. 

X  =  —  39°  34'    1"       .         .         tan.  9.9171381  - 
L  =     101  44  18  ar.  co.  sin.  0.0091788 


y  =     139  50  14         .         .         tan.  9.9263169— 
w  =       23  27  41  


w  =  163  17  55  .  .  cos.  9.9812819 
L  =  101  44  18  .  .  tan.  0.6823798- 
y  =  139  50  14  .  ar.  co.  cos.  0.1 167843 


Right  ascen  =       99  24  48  tan.  0.7804460- 

R  =       992448         .         .         sin.  9.9941121 
-hw  =     163  17  55         .         .         tan.  9.477 1803- 


Dec.=       16  29  20  S.    .      ';.         tan.  9. 47 12924— 

2.  Given  the  longitude  of  Aldebaran  67°  33'  5",  and  latitude 
5°  28'  38"  S.,  and  the  obliquity  of  the  ecliptic  23°  27'  36",  on  the 
1st  of  January,  1840,  to  find  the  right  ascension  and  declination. 

Ans.  Right  ascension  66°  41'  4",  and  declination  16°  10'57"N. 


PROBLEM   XXVI. 

The  Longitude  and  Declination  of  a  Body  being  given,  and  also 
the  Obliquity  of  the  Ecliptic,  to  find  the  Angle  of  Position. 

The  formula  is 

log.  ship  =  log.  sin  w  +  log.  cos  L  +  ar.  co.  log.  cos  D  —  10  : 

p  =  Angle  of  Position  (required) ;       ^ 

L  =  Longitude  ; 

D  =  Declination  ;     „ 

w  =  Obliquity  of  the  ecliptic. 

The  angle  of  position  p  must  be  taken  less  than  90°.  It  is  to  be 
observed  also  that  when  the  longitude  is  less  than  90°,  or  more 
than  270°,  the  northern  part  of  the  circle  of  latitude  lies  to  the  west 
of  the  circle  of  declination,  but  that  when  the  longitude  is  between 
90°  and  270°,  it  lies  to  the  east. 

Note.  The  angle  of  position  may  also  be  computed  from  the 


TO  FIND  THE  TIME  OF  NEW  OR  FULL  MOON  311 

right  ascension  and  latitude,  by  means  of  a  formula  similar  to  thaf 
just  given,  namely, 

log.  ship  =  log.  sin  w  +  log.  cos  R  +  ar.  co.  log.  cos  X  —  10; 
Exam.  1.    Given  the  longitude  of  Regulus  147°  27'  54",  and 
declination  12°  47' 45"  N.,  and  the  obliquity  of  the  ecliptic  23° 
27'  41",  to  find  the  angle  of  position. 

u  =  23°  27' 41"  .  IjSfc  sin.  9.6000260 
L=147  27  54  V  y,*  cos.9.9258601 
D=  12  4745  ar.  co.cos.0.0109217 


Angle  of  pos.  =    20      7  58      [?  ;  ;  <Ji     sin.  9.5368078 

The  circle  of  latitude  lies  to  the  east  of  the  circle  of  declination. 
2.  Given  the  longitude  of  Fomalhaut  331°  27'  56",  and  declina- 
tion 30°  31'  14"  S.,  and  the  obliquity  of  the  ecliptic  23°  27'  41", 
to  find  the  angle  of  position.  Ans.  23°  57'  20". 

The  circle  of  latitude  lies  to  the  west  of  the  circle  of  declination. 


PROBLEM  XXVII. 

To  find  from  the  Tables  the  Time  of  New  or  Full  Moon,  for  a 
given  Year  and  Month. 

For  New  Moon. 

Take  from  Table  LXXXVI,  the  time  of  mean  new  moon  in 
January,  and  the  Arguments  I,  II,  III,  and  IV,  for  the  given  year. 
Take  Irom  Table  LXXXVII,  as  many  lunations  with  the  corre- 
sponding variations  of  Arguments  I,  II,  III,  and  IV,  as  the  given 
month  is  months  past  January,  and  add  these  quantities  to  the  for- 
mer, rejecting  the  ten  thousands  from  the  sums  in  the  columns  of 
the  first  two  arguments,  and  the  hundreds  from  the  sums  in  the 
columns  of  the  other  two.  Seek  the  number  of  days  from  the  first 
of  January  to  the  first  of  the  given  month,  in  the  second  or  third 
column  of  Table  LXXXVIII,  according  as  the  given  year  is  a 
common  or  bissextile  year,  and  subtract  it  from  the  sum  in  the  col- 
umn of  mean  new  moon :  the  remainder  will  be  tabular  time  of 
mean  new  moon  for  the  given  month.  It  will  sometimes  happen 
that  the  number  of  days  taken  from  Table  LXXXVIII,  will  ex- 
ceed the  number  of  days  of  the  sum  in  the  column  of  mean  new 
moon  :  in  this  case  one  lunation  more,  with  the  corresponding  ar- 
guments, must  be  added. 

With  the  sums  in  the  columns  I,  II,  III,  and  IV,  as  arguments, 
take  the  corresponding  equations  from  Table  LXXXIX,  and  add 
them  to  the  time  of  mean  new  moon  :  the  sum  will  be  the  Approxi- 
mate time  of  new  moon  for  the  given  month,  expressed  in  mean 
time  at  Greenwich. 

Next,  for  the  approximate  time  of  new  moon  calculate  the  true 
longitudes  and  hourly  motions  in  longitude  of  the  sun  and  moon ; 


312 


ASTRONOMICAL  PROBLEMS. 


subtract  the  less  longitude  from  the  greater,  and  the  hourly  mo- 
tion of  the  sun  from  the  hourly  motion  of  the  moon  ;  and  say,  as 
the  difference  between  the  hourly  motions  :  the  difference  between 
the  longitudes  :  :  60  minutes  :  the  correction  of  the  approximate 
time.  The  correction  added  to  the  approximate  time,  when  the 
sun's  longitude  is  greater  than  the  moon's,  but  subtracted,  when 
it  is  less,  will  give  the  true  time  of  new  moon  required,  in  mean 
time  at  Greenwich.  This  time  may  be  reduced  to  the  meridian 
of  any  given  place  by  Problem  V. 

For  Full  Moon. 

Take  from  Table  LXXXVI,  the  time  of  mean  new  moon,  and 
the  corresponding  Arguments  I,  II,  III,  and  IV,  for  January  of  the 
given  year,  and  from  Table  LXXXVII,  a  half  lunation  with  the 
corresponding  changes  of  the  arguments.  Then,  when  the  time 
of  mean  new  moon  for  January  is  on  or  after  the  16th,  subtract  the 
latter  quantities  from  the  former,  increasing,  when  necessary  to 
render  the  subtraction  possible,  either  or  both  of  the  first  two  argu 
ments  by  10,000,  and  of  the  last  two  by  100  ;  but  add  them  when 
the  time  is  before  the  16th.  The  result  will  be  the  tabular  time 
of  mean  full  moon  and  the  corresponding  arguments,  for  January. 
Proceed  to  find  the  approximate  time  of  full  moon  after  the  same 
manner  as  directed  for  the  new  moon. 

For  the  approximate  time  of  full  moon  calculate  the  true  longi- 
tudes and  hourly  motions  in  longitude  of  the  sun  and  moon.  Sub- 
tract the  sun's  longitude  from  the  moon's,  adding  360°  to  the  latter 
if  necessary.  Take  the  difference  between  the  remainder  and  VI 
signs,  and  call  the  result  R.  Also  subtract  the  hourly  motion  of 
the  sun  from  the  hourly  motion  of  the  moon.  Then  say,  as  the 
difference  between  the  hourly  motions  :  R  :  :  60m.  :  the  correction 
of  the  approximate  time.  The  correction  added  to  the  approxi- 
mate time  of  full  moon,  when  the  excess  of  the  moon's  longitude 
over  the  sun's  is  less  than  VI  signs,  but  subtracted  when  it  is 
greater,  will  give  the  true  time  of  full  moon. 

Exam.  1.  Required  the  time  of  new  moon  in  September,  1838, 
expressed  in  mean  time  at  New  York. 


1838, 
81un. 

M.  New  Moon. 

I. 

n. 

m. 

IV. 

d.   h.   m. 

24  16  53 
236   5  52 

0681 
6468 

9175 
5737 

99 
22 

85 
93 

Dap, 

260  22  45 
243 

7149  4912   21 
Approximate  time. 

78 

Sept'r, 

II. 
III. 
IV. 

17  22  45 
0  16 
9  35 
3 
10 

Sept'r. 

18   8  49 

TO  FIND  THE  TIME  OF  NEW  OR  FULL  MOON. 


313 


Moon's  true  long,  found  for  approx.  time,  is     5s-  25°  29'  19" 
Sun's  do.  do.  do.  5     25    27    27 

Difference,         .        y^     •     ~:' ••''•    '•&£.' 


5     25    27    27 
1    52 


Moon's  hourly  motion  in  long,  is  29    28 

Sun's  do.  do.         /.;    '.,-*'       .  2    27 

Difference,         ;''*   :.         ."      ,•  .  '      .     '*./"'  27      1 

As  27'  I"  :  I'  52"  : :  60m- :  4m-  98-,  the  correction. 

Approx.  time  of  new  moon,  September,         .      18d-  8h>  49m-  0" 
Correction,         .  "    "'*f*      .      *V1j     .         .  —4     9 

True  time,  in  mean  time  at  Greenwich,        .      18    8    44  51 
Diff.  of  meridians,       .         .         .,.-*<.•  4    56     4 

True  time,  in  mean  time  at  New  York,         .      18    3    48  47 

Exam.  2.  Required  the  time  of  full  moon  in  April,  1838,  ex 
pressed  in  mean  time  at  New  York. 


1838, 

ilun. 

M.  Full  Moon. 

I. 

II. 

III. 

rv. 

d.       h.       m. 
24     16     53 
14    18    22 

0681 
404 

9175 
5359 

99 

58 

85 
50 

3  lun. 

9     22    31 

88     14    12 

0277 
2425 

3816 
2151 

41 
46 

35 

97 

Days, 

98     12    43 
90 

2702     5967       87 
Approximate  time. 

32 

April, 

II. 
III. 
IV. 

8    12    43 
8    29 
16      7 
15 
30 

April, 

9     14      4 

Moon's  true  long,  found  for  approx.  time,  is     6s-  19°  44'  17;/ 


Sun's 


do. 


do. 


do. 


0     19    45    22 


29    58    55 
000 


R.     .     1  5 

Moon's  hourly  motion  in  long,  is          .                          30  15 

Sun's             do.             do.                    .                            2  27 

Difference .   27  48 

As  27'  48"  :  1'  5"  : :  60m-  :  2m-  208-,  the  correction. 

40 


314  ASTRONOMICAL  PROBLEMS. 

Approximate  time  of  full  moon,  April,     '* •''?!**    9d-  14ht  4m>  0s- 
Correction,     •%         .      *»P        .         .         .  +  2  20 


True  time,  in  mean  time  at  Greenwich,       .        9    14     6  20 
Diff.  of  meridians,     .....  4  56     4 


True  time,  in  mean  time  at  New  York,       .        9      91016 

3.  Required  the  time  of  new  moon  in  September,  1837,  ex- 
pressed in  mean  time  at  Philadelphia ;  taking  the  longitudes  for 
the  approximate  time  from  the  Nautical  Almanac. 

Ans.  29d.  3h.  Om.  5s. 

4.  Required  the  time  of  full  moon,  in  October,  1837,  expressed 
in  mean  time  at  Boston.  Ans.  13d.  6h.  30m.  25s. 


PROBLEM  XXVIII. 

To  determine  the  number  of  Eclipses  of  the  Sun  and  Moon  that 
may  be  expected  to  occur  in  any  given  Year,  and  the  Times 
nearly  at  which  they  will  take  place. 

For  the  Eclipses  of  the  Sun. 

Take,  for  the  given  year,  from  Table  LXXXVI  the  time  of 
mean  new  moon  in  January,  the  arguments  and  the  number  N. 
If  the  number  N  differs  less  than  37  from  either  0,  500,  or  1000, 
an  eclipse  must  occur  at  that  new  moon.  If  the  difference  is  be- 
tween 37  and  53,  there  may  be  an  eclipse,  but  it  is  doubtful,  and 
the  doubt  can  only  be  removed  by  a  calculation  of  the  true  places 
of  the  moon  and  sun.  If  the  difference  exceeds  53,  an  eclipse  is 
impossible. 

If  an  eclipse  may  or  must  occur  at  the  new  moon  in  January, 
calculate  the  approximate  time  of  new  moon  by  Problem  XXVII, 
and  it  will  be  the  time  nearly  of  the  middle  of  the  eclipse,  express- 
ed in  mean  time  at  Greenwich.  This  may  be  reduced  to  the 
meridian  of  any  other  place  by  Problem  V. 

To  find  the  first  new  moon  after  January,  at  which  an  eclipse 
of  the  sun  maybe  expected,  seek  in  column  N  of  Table  LXXXVII 
the.  first  number  after  that  answering  to  the  half  lunation,  that, 
added  to  the  number  N  for  the  given  year,  will  make  the  sum  come 
within  53  of  0,  500,  or  1000.  Take  the  corresponding  lunations, 
changes  of  the  arguments,  and  the  number  N,  and  add  them,  re- 
spectively, to  the  mean  new  moon  in  January,  the  arguments,  and 
the  number  N,  for  the  given  year.  Take  from  the  second  or  third 
column  of  Table  LXXXVIII,  according  as  the  given  year  is  a 
common  or  bissextile  year,  the  number  of  days  next  less  than  the 
days  of  the  sum  in  the  column  of  mean  new  moon,  and  subtract  it 
from  this  sum ;  the  remainder  will  be  the  tabular  time  of  mean 
new  moon  in  the  month  corresponding  to  the  days  taken  from  Ta- 


TO  FIND  THE  NUMBER  OF  ECLIPSES  IN  A  YEAR. 


315 


ble  LXXXVIII.  At  this  new  moon  there  may  be  an  eclipse  of 
the  sun ;  and  if  the  sum  in  the  column  N  is  within  37  of  the  num- 
bers mentioned  above,  there  must  be  one.  Find  the  approximate 
time  of  new  moon,  and  it  will  be  the  time  nearly  of  the  middle  of 
the  eclipse. 

If  any  of  the  other  numbers  in  the  last  column  of  Table 
LXXXvII  are  found,  when  added  to  the  number  N  of  the  given 
year,  to  give  a  sum  that  falls  within  the  limit  53,  proceed  in  a  simi- 
lar manner  to  find  the  approximate  times  of  the  eclipses. 

Note.  When  the  sum  of  the  numbers  N,  or  the  number  N  itself, 
in  case  the  eclipse  happens  in  January,  is  a  little  above  0,  or  a 
little  less  than  500,  the  moon  will  be  to  the  north  of  the  sun,  and 
there  is  a  probability  that  the  eclipse  will  be  visible  at  any  given 
place  in  north  latitude  at  which  the  approximate  time  of  the  eclipse, 
found  as  just  explained  and  reduced  to  the  meridian  of  the  place, 
comes  during  the  day-time.  When  the  number  N  found  for  the 
eclipse  is  more  than  500,  the  moon  will  be  to  the  south  of  the  sun, 
and  the  eclipse  will  seldom  be  visible  in  the  northern  hemisphere, 
except  near  the  equator. 

For  the  Eclipses  of  the  Moon. 

Find  the  time  of  full  moon  and  the  corresponding  arguments  and 
number  N,  for  January  of  the  given  year,  as  explained  in  Problem 
XXVII.  Then  proceed  to  find  the  times  at  which  eclipses  of  the 
moon  may  or  must  occur,  after  the  same  manner  as  for  eclipses  of 
the  sun,  only  making  use  of  the  limits  35  and  25,  instead  of  53 
and  37.* 

Note.  An  eclipse  of  the  moon  will  be  visible  at  a  given  place, 
if  the  time  of  the  eclipse  thus  found  nearly,  and  reduced  to  the 
meridian  of  the  place,  comes  in  the  night. 

Exam.  1 .  Required  the  eclipses  that  may  be  expected  in  the 
year  1840,  and  the  times  nearly  at  which  they  will  take  place. 

For  the  Eclipses  of  the  Sun. 


1840, 
21un. 

M.  New  Moon. 

I. 

II. 

III. 

rv. 

N. 

d.      h.      m. 
3     10     30 
59       1     28 

0085 
1617 

6386 
1434 

65 
31 

63 

98 

844 
170 

62     11     58 
60 

1702     7820       96         61        014 

As   the  sum  of  the  numbers  N 
comes  within  37  of  0,  there  must  be 
an  eclipse. 

• 
Mean  time  at  Greenwich. 

March, 
I. 
II. 
III. 
IV. 

2    11    58 
8      3 
19    38 
12 
13 

March, 

3     16      4 

*  The  numbers  53,  37,  and  35,  25,  are  the  lunar  and  solar  ecliptic  limits,  as 
determined  by  Delambre.  The  limits  given  in  the  text,  converted  into  thousandth 
parts  of  the  circle,  are  55,  37,  and  37,  21. 


816 


ASTRONOMICAL  PROBLEMS. 


1840, 
8  Ion. 

M.  New  Moon. 

I. 

II. 

III. 

IV. 

N. 

d.        h.      m. 
3     10    30 
236      5    52 

0085 
6468 

6386 
5737 

65 
22 

63 
93 

844 
682 

239     16    22 
213 

6553      2123       87         56        526 

As  the  sum  of  the  numbers  N 
comes  within  37  of  500,  there  must 
be  an  eclipse. 

Mean  time  at  Greenwich 

August, 

II*. 
III. 
IV. 

26    16    22 
0    54 
0    49 
15 
16 

August, 

26    18    36 

For  the  Eclipses  of  the  Moon. 


1840, 

ilun. 

M.  Full  Moon. 

I. 

II. 

m. 

IV. 

N. 

3     10    30 
14     18    22 

0085 
404 

6386 
5359 

65 

58 

63 

50 

844 
43 

llun. 

18      4    52 
29     12    44 

489 
808 

1745 
717 

23 
15 

13 
99 

887 
85 

47     17    36 
31 

1297     2462       38         12        972 

As  the  sum  of  the  numbers  N,  al- 
though it  comes  within  35  of  1000, 
does  not  come  within  25,  the  eclipse 
may  be  considered  doubtful. 

Mean  time  at  Greenwich. 

Febr. 
I. 
II. 
III. 
IV. 

16    17    36 
7    27 
0    23 
5 
27 

Febr. 

17      1    58 

1840, 
7  lun. 

M.  Full  Moon. 

I. 

H. 

III. 

IV. 

N. 

d.       h.       m. 
18      4    52 
206     17      8 

489 
5659 

1745 

5020 

23 

7 

13 

94 

887 
596 

224    22      0 
213 

6148      6765       30         07        483 

As  the  sum  of  the  numbers   N 
comes  within  25  of  500,  there  must 
be  an  eclipse. 

Mean  time  at  Greenwich. 

August, 

IL 
III. 
IV. 

11    22      0 
1    37 
19    16 
3 
25 

August, 

12    19    21 

2.  Required  the  eclipses  that  may  be  expected  in  the  year  1839, 
and  the  times  nearly  at  which  they  will  take  place,  expressed  in 
mean  civil  time  at  New  York. 


TO  CALCULATE  A  LUNAR  ECLIPSE.  317 

Ans.  One  of  the  sun  on  the  15th  of  March,  at  9h.  20m.  A.  M. ; 
and  one  of  the  sun  on  the  7th  of  September,  at  5h.  24m.  P.  M. 

3.  Required  the  eclipses  that  may  be  expected  in  the  year  1841, 
and  the  times  nearly  at  which  they  will  take  place,  expressed  in 
mean  civil  time  at  New  York. 

Ans.  Four  of  the  sun,  namely,  one  on  the  22d  of  January,  at 
12h.  18m.  P.  M. ;  one  on  the  21st  of  February,  at6h.  17m.  A.M. ; 
one  on  the  18th  of  July,  at  9h.  24m.  A.  M. ;  and  one  on  the  16th 
of  August,  at  4h.  28m.  P.M.:  and  two  of  the  moon,  namely,  one 
on  the  5th  of  February,  at  9h.  10m.  P.  M. ;  and  one  on  the  2d  of 
August,  at  5h.  5m.  A.  M. 

The  eclipses  of  the  sun  in  January  and  August  may  be  con- 
sidered as  doubtful. 


PROBLEM   XXIX. 

To  calculate  an  Eclipse  of  the  Moon. 

The  calculation  of  the  circumstances  of  a  lunar  eclipse  is  effect- 
ed with  the  following  fundamental  data,  derived  from  the  tables  of 
the  sun  and  moon  : 

Approximate  Time  of  Full  Moon  (at  Greenwich),  T 

Sun's  Longitude  at  that  time,  L 

Do.  Hourly  Motion, s 

Do.  Semi-diameter, <5 

Do.  Parallax, p 

Moon's  Longitude, I 

Do.      Latitude, X 

Do.      Equatorial  Parallax,  P 

Do.      Semi-diameter, d 

Do.      Hourly  Motion  in  longitude,  m 

Do.      Hourly  Motion  in  latitude,  n 

We  obtain  the  time  T  by  Problem  XXVII ;  the  quantities  ap- 
pertaining to  the  sun,  namely,  L,  s,  and  5,  by  Problem  IX  ;*  and 
those  which  have  relation  to  the  moon,  namely,  /,  X,  P,  d,  m,  and 
n,  by  Problem  XIV. 

From  these  quantities  we  derive  the  following : 

True  Time  of  Full  Moon,  (at  given  place,)  .  T' 

Moon's  Latitude  at  that  time,    .         .  .  X' 

Semi-diameter  of  earth's  shadow,       .  .  S 

Inclination  of  Moon's  relative  orbit,    .         .  ;  r  I 

T  being  known,  T'  is  found  as  explained  in  Problem  XXVII. 
To  obtain  X',  we  state  the  following  proportion, 

1  hour :  correction  for  the  time  of  full  moon  ::»:*; 
*  p  may  be  taken  =  9". 


318  ASTRONOMICAL  PROBLEMS. 

from  this  we  deduce  the  value  of  x  ;  and  thence  find  X  by  the 
equation 

V  =  X  ±  x. 

When  the  true  time  of  full  moon,  expressed  in  mean  time  at 
Greenwich,  is  later  than  the  approximate  time,  the  upper  sign  is 
to  be  used,  if  the  latitude  is  increasing,  the  lower  if  it  is  decreas- 
ing ;  but  when  the  true  time  is  earlier  than  the  approximate  time, 
the  lower  sign  is  to  be  used  if  the  latitude  is  increasing  ;  the  upper 
if  it  is  decreasing. 

The  value  of  S  is  derived  from  the  equation 


and  the  angle  I  from  the  formula 

log.  tang  I  =  log.  n  +  ar.  co.  log.  (m  —  s). 
The  foregoing  quantities  having  all  been  determined,  the  various 
circumstances  of  the  eclipse  may  be  calculated  by  the  following 
formulae  : 

For  the  Time  of  the  Middle  of  the  Eclipse. 

3.55630  +  log.  cos  I  +  ar.  co.  log.  (m  —  s)  —  20  =  R  ; 

log.  t  =  R  -f  log.  X'  +  log.  sin  I  —  10  ; 

M  =  T'  ±  t  : 

t  =  interval  between  time  of  middle  of  eclipse  and  time  of  full 
moon  ;  M  =  time  of  middle  of  the  eclipse. 

The  upper  sign  is  to  be  taken  in  the  last  equation  when  the  lati- 
tude is  decreasing;  the  lower,  when  it  is  increasing. 

For  the  Times  of  Beginning  and  End. 

log.  c  =  log  X'  +  log.  cos  I  —  10  ; 
log  >0^log.(S  +  «*+c)  +  log.(S+d--c)  {  R. 

B  =  M  —  v,  and  E  =  M  +  v  : 

v  —  half  duration  of  the  eclipse  ;  B  =  time  of  beginning  ;  and  E  = 
time  of  end. 

Note.  If  c  is  equal  to  or  greater  than  S  +  d,  there  cannot  be  an 
eclipse. 

For  the  Times  of  Beginning  and  End  of  the  Total  Eclipse. 


R  . 


B'  =  M  —  v',  and  E'  =  M  +  v1  : 

v'  =  half  duration  of  the  total  eclipse  ;  B'  =  time  of  beginning  of 
total  eclipse  ;  and  E'  =  time  of  end  of  total  eclipse. 

Note.  When  c  is  greater  than  S  —  d  ,  the  eclipse  cannot  be  total. 

For  the  Quantity  of  the  Eclipse. 

log.  Q  =  0.77815  +  log.  (S  +  d  -  c)  +  ar.  co.  log.  d  -  10  ; 
Q  =  the  quantity  of  the  eclipse  in  digits. 


TO  CALCULATE  A  LUNAR  ECLIPSE.  319 

Note  1  .  An  eclipse  of  the  moon  begins  on  the  eastern  limb,  and 
ends  on  the  western.  In  partial  eclipses  the  southern  part  of  the 
moon  is  eclipsed  when  the  latitude  is  north,  and  the  northern  part 
when  the  latitude  is  south. 

Note  2.  When  the  eclipse  commences  before  sunset,  and  ends 
after  sunset,  the  moon  will  rise  more  or  less  eclipsed."  To  obtain 
the  quantity  of  the  eclipse  at  the  time  of  the  moon's  rising,  find 
the  moon's  hourly  motion  on  the  relative  orbit  by  the  equation 

log.  h  —  log.  (m  —  s)  +  ar.  co.  log.  cos  I  ; 

in  which  h  =  hourly  motion  on  relative  orbit.  Also  find  the  inter- 
val between  the  time  of  sunset  and  the  time  of  the  middle  of  the 
eclipse,  which  call  i.  Then, 

1  hour  :  i  :  :  h  :  x. 

Deduce  the  value  of  x  from  this  proportion,  and  substitute  it  in 
the  equation 


in  which  c  designates  the  same  quantity  as  in  previous  formulae. 
Find  the  value  of  c',  and  use  it  in  place  of  c  in  the  above  formula 
for  the  quantity  of  the  eclipse,  and  it  will  give  the  quantity  of  the 
eclipse  at  the  time  of  the-  moon's  rising.  When  the  eclipse  begins 
before  and  ends  after  sunrise,  the  quantity  of  the  eclipse  at  the 
time  of  the  moon's  setting  may  be  found  in  the  same  manner,  only 
using  sunrise  instead  of  sunset. 

Example.  Required  to  calculate,  for  the  meridian  of  New  York, 
the  eclipse  of  the  moon  in  October,  1837. 


Elements. 


Approximate  time  of  full  moon, 
Sun's  longitude  at  that  time,     . 

Do.  hourly  motion 

Do.  semi-diameter 

Do.  parallax, 
Moon's  longitude, 

Do.  latitude, 

Do.  equatorial  parallax, 

Do.  semi-diameter, 

Do.  hourly  motion  in  long. 


T  =llh-  10m-(0ct.  13) 

L  ==  6s-  20°  24'  28/; 
s   =  2  29 

*   =  16     4 

=  9 

=  0    20    21   51 
X  =  11   28  S. 

P  =  59   32 

d  =  16   13 

m=  35  54 


Do.  hourly  motion  in  lat.  (tending  north),  n  =  3   19 

Approx.  time  of  full  moon,  October,          .         13d<  llh-  10m-  00"- 
Correction  found  by  Prob.  XXVII,  .  +4     42 

True  time,  in  mean  time  at  Greenwich,    .         13    11     14     42 
Diff.  of  meridians, 4    56       4 


True  time,  in  mean  time  at  New  York,  T'  =    13     6    18     38 


320 


ASTRONOMICAL   PROBLEMS. 


60m-  :  41 
Moon's  lat.  at  approx.  time, 
Correction,  .      4  ±*& 

Moon's  lat.  at  true  time, 

Moon's  equatorial  parallax, 
Sun's  do 

Sum, 

Sun's  semi-diameter,    . 

Diff.  . 

Add 


Semi-diameter  of  earth's  shadow,  .         .         . 

Moon's  hor.  mot.  less  sun's  (m  —  s)  =  2005"  .  ar.  co 
Moon's  hor.  motion  in  latitude,      n  =    199    .         . 

Inclination  of  rel.  orbit,  I  =  5°  40'          .         .         . 
Time  of  Middle. 


=  11'  28"  S. 
=  —16 


59  41 

5=164 


P+p  — 


S  =  44  21 

log.  6.69789 
log.  2.29885 

tan.  8.99674 


3.5563C 

5°  40'     .         .         cos.  9.99787 
2005"  ar.  co.  log.  6.69789 


V        . 
I 

t 

T'       . 

Middle, 


X' 
I 


S-fd 


672" 


5°  40' 


Oh-    lra-  588-  = 
6    18     38P.M. 


R.  0.25206 
log.  2.82737 
sin.  8.99450 

log.  2.07393 


.  6.20     36P.M. 

Times  of  Beginning  and  End. 


11'  9"  =  669" 

4303" 
2965 


lh-46m«  22'- =  6382* 


log.  2.82737 
cos.  9.99787 

log.  2.82524 

log.  3.63377 
log.  3.47202 

2  )  7.10579 

3.55289 
R.  0.25206 

log.  3.80495 


Middle, 

Beginning, . 
Ei 


TO  CALCULATE  A  SOLAR  ECLIPSE  321 

lh-  46m-  22s-  =  6382s-    .    log.  3.80495 
6  20  36 


4  34  14  P.  M. 
8   6  58  P.  M. 


S  -d-c 


Middle, 


2357" 
1019 


Oh.  46m.  9«.=  2769s- 
6  20  36 


Beg.  of  total  eclipse,    5    34     27  P.  M. 
End  of  total  eclipse,    7      6     45  P.  M 


S+d-c 
d 

Quantity, 


log.  3.37236 
log.  3.00817 

2  )  6.38053 

3.19026 
R     0.25206 


log.  3.44232 


0.77815 

log.  3.47202 

973"  .      ar.  co.  log.  7.01189 


18.3  digits, 


log.  1.26206 


PROBLEM  XXX. 

To  calculate  an  Eclipse  of  the  Sun,  for  a  given  Place. 

Having  found  by  the  rule  given  in  the  note  to  Problem  XXVIII, 
that  there  is  a  probability  that  the  eclipse  will  be  visible  at  the 
given  place,  and  calculated  the  approximate  time  of  new  moon  by 
Problem  XXVII,  find  from  the  tables,  for  this  time  or  for  the  near- 
est whole  or  half  hour,  the  sun's  longitude,  hourly  motion,  and 
semi-diameter  ;  and  the  moon's  longitude,  latitude,  equatorial  par- 
allax, semi-diameter,  and  hourly  motions  in  longitude  and  latitude. 
Find  also  by  Problem  XVI,  the  longitude  and  altitude  of  the 
nonagesimal  degree  ;  and  thence  compute  by  Problem  XVII,  the 
apparent  longitude,  latitude,  and  augmented  semi-diameter  of  the 
moon,  (using  the  relative  horizontal  parallax.)  With  these  data 
compute  the  apparent  distance  of  the  centres  of  the  sun  and 
at  the  time  in  question,  by  means  of  the  following  formulae : 

log.  tang  &  =  log.  X'  +  ar.  co.  log.  a  ; 
log.  A  =  log.  a  +  ar.  co.  log.  cos  6  : 
41 


ASTRONOMICAL    PROBLEMS. 

in  which 

A  =  appar.  distance  of  centres  ; 
X'  =  appar.  Lat.  of  Moon ; 

a  =  Dift*.  of  appar.  Long,  of  Moon  and  Sun  =  diff.  of  appar 
long,  of  Moon  (found  as  above)  and  true  long,  of  Sua 

6  is  an  auxiliary  arc.  The  value  of  6  being  derived  from  th« 
first  equation,  the  second  will  then  make  known  the  value  of  A. 

a  and  X'  are  in  every  instance  to  be  affected  with  the  positive 
sign.* 

For  the  Approximate  Times  of  Beginning,  Greatest  Obscuration, 

and  End. 

Let  the  time  for  which  the  above  calculations  are  made,  be  de- 
noted by  T.  If  the  apparent  distance  of  the  centres  of  the  sun 
and  moon,  found  for  the  time  T,  is  less  than  the  sum  of  their  ap- 
parent semi-diameters,  there  is  an  eclipse  at  this  time.  But  if  it 
is  greater,  either  the  eclipse  has  not  yet  commenced,  or  it  has  al- 
ready terminated.  It  has  not  commenced  if  the  apparent  longitude 
of  the  moon  is  less  than  the  longitude  of  the  sun ;  and  has  termi- 
nated, if  the  apparent  longitude  of  the  moon  is  greater  than  the 
longitude  of  the  sun. 

1.  If  there  should  be  an  eclipse  at  the  time  T,  from  the  sun's 
longitude  and  hourly  motion  in  longitude,  and  the  moon's  longi- 
tude and  latitude,  and  hourly  motions  in  longitude  and  latitude, 
found  for  this  time,  calculate  the  longitudes  and  the  moon's  lati- 
tude for  two  instants  respectively  an  hour  before,  and  an  hour  after 
the  time  T.  The  semi-diameter  of  the  sun,  and  the  equatorial 
parallax  and  semi-diameter  of  the  moon,  may,  in  our  present  in- 
quiry, be  regarded  as  remaining  the  same  during  the  eclipse.  Find 
the  apparent  longitude  and  latitude,  and  the  augmented  semi-diam- 
eter of  the  moon,  (using  in  all  cases  the  relative  parallax,)  and 
thence  compute  by  the  formulae  already  given,  the  apparent  dis 
tance  of  the  centres  of  the  sun  and  moon  at  the  two  instants  in 
question. 

Observe  for  each  result,  whether  it  is  less  or  greater  than  the 
sum  of  the  apparent  semi-diameters  of  the  two  bodies.  If  the 
moon  is  apparently  on  the  same  side  of  the  sun  at  the  times  T  and 
T  +  lh.,  take  the  difference  of  the  distances  of  the  two  bodies  in 
apparent  longitude  at  these  times,  but,  if  it  is  on  opposite  sides, 
take  their  sum,  and  it  will  be  the  variation  of  this  distance  in  the 

*  A,  the  apparent  distance  of  the  centres,  may  be  found  without  the  aid  of  loga- 
rithms by  means  of  the  following  equation : 

A  =  V  a*  4-  A'2. 

If  the  logarithmic  formulae  are  used,  it  will  be  sufficient  here  to  take  out  the  angle 
0  to  the  nearest  minute.  When  we  have  occasion  to  obtain  the  distance  of  the 
centres  exact  to  within  a  small  fraction  of  a  second,  0  must  be  taken  to  the  nearest 
tens  of  seconds,  if  it  exceeds  20°  or  30°. 


TO  CALCULATE  A  SOLAR  ECLIPSE.  323 

hour  following  T.  Find  in  like  manner  the  variation  of  the  dis» 
tance  during  the  hour  preceding  T.  Then,  if  the  apparent  distance 
of  the  centres  at  the  times  (T  —  lh.),  (T  +  lh.)  is  less  than  the 
sum  of  the  apparent  semi-diameters,  deduce  from  these  results 
the  variations  of  the  distance  in  apparent  longitude  during  the  pre- 
ceding and  following  hours,  allowing  for  the  second  difference,  and 
observing  whether  the  two  bodies  are  approaching  each  other,  or 
receding  from  each  other.  Thence,  find  the  distance  in  apparent 
longitude  at  the  times. (T  —  2h.),  (T  +  2h.)  Find  by  the  same 
method  the  apparent  latitude  of  the  moon  at  the  instants  ( T  —  2h.), 
(T  +  2h.),  observing  that  the  variation  of  the  apparent  latitude  in 
any  given  interval  is  the  difference  between  the  latitudes  at  the 
beginning  and  end  of  it,  if  they  are  both  of  the  same  name  ;  their 
sum,  if  they  are  of  opposite  names. 

From  these  results  derive  the  apparent  distance  of  the  centres 
of  the  sun  and  moon  at  the  two  instants  in  question. 

If  there  should  still  be  an  eclipse  at  the  time  (T  -f-  2h.)  or 
(T  —  2h.),  find  by  the  same  method  the  distance  of  the  centres  at 
the  time'  (T  +  3h.)  or  (T  -  3h.)  These  calculations  being  effect- 
ed, the  times  of  the  beginning,  greatest  obscuration,  and  end  of  the 
eclipse,  will  fall  between  some  of  the  instants  T,(T—  lh.),(T  +  lh.), 
&c.,  for  which  the  apparent  distance  of  the  centres  is  computed. 

2.  If  the  eclipse  occurs  after  the  time  T,  the  different  phases 
will  happen  between  the  instants  T,  (T  +  lh.),  (T  +  2h.),  &c. 
Find  the  apparent  distance  of  the  centres  of  the  sun  and  moon  for 
the  times  (T  +  lh.),  (T  +  2h.),  by  the  same  method  as  that  by 
which  it  is  found  for  the  times  (T  +  lh.),  (T  —  lh.),  in  the  case 
just  considered.     Then,  if  the  eclipse  has  not  terminated,  deduce 
the  distance  of  the  moon  from  the  sun  in  apparent  longitude,  and 
the  moon's  apparent  latitude,  for  the  time  (T  +  3h.),  from  these 
distances  and  latitudes  at  the  times  T,  (T  +  lh.),  (T  +  2h.) ;  as 
in  the   preceding   case  the   distance   and  latitude  for  the   time 
(T+2h.)  were  deduced  from  the  same  at  the  times  (T  —  lh.),  T, 
(T-j-lh.)     With  the  results  obtained  compute  the  apparent  dis- 
tance of  the  centres  of  the  two  bodies  at  the  time  (T  -f  3h.) 

3.  In  case  the  eclipse  occurs  before  the  time  T,  the  apparent 
distance  of  the  centres  must  be  found  by  similar  methods  for  the 
times  (T  -  lh.),  (T  -  2h.),  &c. 

The  calculation  is  to  be  continued  until  the  distance,  from  being 
less,  becomes  greater  than  the  sum  of  the  semi-diameters. 

Now,  let  h  =  variation  of  apparent  distance  of  centres  in  the 
interval  of  one  hour  comprised  between  the  first  two  of  the  instants 
for  which  the  distance  is  computed ;  d  —  difference  between  the 
sum  of  the  semi-diameters  of  the  sun  and  moon  and  the  apparent 
distance  of  their  centres  at  the  first  instant ;  and  t  =  interval  be- 
tween first  instant  and  the  time  of  the  beginning  of  the  eclipse. 
Then, 

h  :  d  :  :  60m-  -  t  (nearly.) 


324  \STRONOMICAL  PROBLEMS. 

Find  the  value  of  t  given  by  this  proportion,  and  add  it  to  the 
time  at  the  first  instant,  and  the  result  will  be  a  first  approximation 
to  the  time  of  the  beginning  of  the  eclipse,  which  call  b.  Find, 
by  interpolation,*  the  distance  of  the  moon  from  the  sun  in  appa- 
rent longitude  (a\  and  the  moon's  apparent  latitude  (V),  for  this 
time,  and  thence  compute  the  apparent  distance  of  the  centres. 
Take  h  =  variation  of  apparent  distance  in  the  interval  between  the 
time  b  and  the  nearest  of  the  two  instants  above  mentioned,  be- 
tween which  the  beginning  falls,  and  d  =  difference  between  the 
apparent  distance  of  the  centres  at  the  time  b  and  the  sum  of  the 
semi-diameters,  and  compute  again  the  value  of  t.  Add  this  to  the 
time  b,  or  subtract  it  from  it,  according  as  b  is  before  or  after  the 
beginning,  and  the  result  will  be  a  second  approximation  to  the 
time  of  the  beginning,  which  call  B.  A  result  still  more  approxi- 
mate may  be  had,  by  taking  h  =  variation  of  apparent  distance  of 
centres  in  the  interval  B  —  b,  d  =  difference  between  apparent  dis- 
tance at  the  time  B  and  sum  of  semi-diameters,  finding  anew  the 
value  of  t  given  by  the  preceding  proportion,  and  adding  it  to,  or 
subtracting  it  from,  as  the  case  may  be,  the  time  B.  But  pfepara- 
tory  to  the  calculation  of  the  exact  times,  it  will  suffice,  in  general, 
to  take  the  first  approximation. 

The  end  of  the  eclipse  will  fall  between  the  last  two  of  the 
several  instants  for  which  the  apparent  distance  of  the  centres  of 
the  moon  and  sun  have  been  computed.  The  approximate  time 
of  the  end  is  found  by  the  same  method  as  that  of  the  beginning.! 

*  The  second  differences  may  easily  be  taken  into  the  account  in  finding  the 
quantities  a  and  A'  for  the  time  6.  Thus,  let  k  =  variation  of  a  for  the  interval  of 
an  hour  comprised  between  the  instants  above  mentioned,  k'  =  same  for  the  suc- 
ceeding hour,  and  i  =  interval  between  b  and  the  nearer  of  the  two  instants,  (in 

£  £ fc' 

minutes.;     Then,  if  we  put  /=  — ,  c  =  — — — ,  and  v  =  var.  of  a  in  interval  z, 

o  ob 


10 

The  upper  sign  is  to  be  used  when  the  time  b  is  nearer  the  first  than  the  second 
instant,  the  lower  when  it  is  nearer  the  second  than  the  first,  c  is  to  be  used  with 
its  sign.  The  error  by  this  method  will  not  exceed  the  number  c,  (supposing  the 
changes  of  k,  k1,  from  10m.  to  10m.  to  increase  or  decrease  by  equal  degrees.) 

•  The    general    formula    for    interpolation    is    Q  =  q  -f-  -  d'  -\  --  -—  ^  —  d"  + 

-  ^—^  "73  -  d'"  -f-  &c.,  in  which  q  is  the  first  of  a  series  of  values,  found  at 

equal  intervals,  of  the  quantity  whose  value  Q  at  the  time  t  is  sought,  t  is  reck- 
oned from  the  time  for  which  q  is  found,  h  is  one  of  the  equal  intervals,  d',  d"f 
d'",  &c.,  are  the  first,  second,  third,  &c.,  differences.  If  we  make  h  =  1,  we  have 


.  ,+.  *+ 


t  In  effecting  the  reductions  of  the  quantities  a  and  V  to  the  first  approximate 
time  of  end,  k1  must  stand  for  the  variation  of  a  during  the  hour  preceding  that 
comprised  between  the  last  two  instants,  and  the  last  instant  must  be  substituted 
for  the  first.  (See  Note  above.) 


TO  CALCULATE  A  SOLAR  ECLIPSE. 


325 


The  middle  of  the  interval  between  the  approximate  times  of 
the  beginning  and  end  of  the  eclipse,  will  be  a  first  approximation 
to  the  time  of  greatest  obscuration. 

Note.  When  the  object  is  merely  to  prepare  for  an  observation, 
results  sufficiently  near  the  truth  may  be  obtained  by  a  graphical 
construction.  The  elements  of  the  construction  are  the  difference 
of  the  apparent  longitudes  of  the  moon  and  sun,  and  the  apparent 
latitude  of  the  moon,  found  as  above,  for  two  or  more  instants  du- 
ring the  continuance  of  the  eclipse.  Draw  a  right  line  EF,  (Fig. 
123,)  to  represent  the  ecliptic,  assume  on  it  some  point  C  for  the 

Fig.  123. 


position  of  the  -sun  at  the  instant  of  apparent  conjunction,  and  lay 
off  CA,  CA',  equal  to  the  two  differences  of  apparent  longitude  ; 
and  to  the  right  or  left,  according  as  the  moon  is  to  the  west  or 
east  of  the  sun  at  the  instants  for  which  the  calculations  have  been 
made.  Erect  the  perpendiculars  Ap,  A'p',  and  mark  off  Aa,  A.' a' 
equal  to  the  two  apparent  latitudes.  Through  a,  a',  draw  a  right 
line,  and  it  will  be  the  apparent  relative  orbit  of  the  moon,  or 
will  differ  but  little  from  it.  From  C  let  fall  the  perpendicular  Cm 
upon  the  relative  orbit,  m  will  be  the  apparent  place  of  the  moon 
at  the  instant  of  greatest  obscuration.  Take  a  distance  in  the  di- 
viders equal  to  the  sum  of  the  apparent  semi-diameters  of  the  moon 
and  sun,  and  placing  one  foot  of  it  at  C,  mark  off  with  the  other 
the  points  /,  f,  for  the  beginning  and  end  of  the  eclipse,  and  by 
means  of  a  square  mark  on  EF  the  points  6,  e,  which  answer  to 
the  beginning  and  end.  If  the  eclipse  be  total  or  annular,  mark 
the  points  of  immersion  and  emersion,  g,  g'y  with  an  opening  in 
the  dividers  equal  to  the  difference  of  the  semi-diameters,  and  find 
the  corresponding  points  £',  e'  on  the  line  EF. 

If  the  calculations  are  made  from  hour  to  hour,  the  distance  AA' 
is  the  apparent  relative  hourly  motion  of  the  sun  and  moon  in  lon- 
gitude. This  distance  laid  off  repeatedly  to  the  right  and  left  will 
determine  the  points  1,  2,  &c.,  answering  to  lh.,  2h.,  &c.  before 


326  ASTRONOMICAL  PROBLEMS. 

and  after  the  times  for  which  the  calculations  are  made.  If  the 
spaces  in  which  the  points  b,  e,  answering  to  the  beginning  and 
end  of  the  eclipse,  occur,  be  divided  into  quarters,  and  then  sub- 
divided into  three  equal  parts  or  five-minute  spaces,  the  approxi- 
mate times  of  the  beginning  and  end  of  the  eclipse  will  become 
known. 

From  the  point  m,  as  a  centre,  describe  the  lunar  disc ;  and 
from  the  point  C,  as  a  centre,  describe  the  sun's  disc,  and  we  shall 
have  the  figure  of  the  greatest  eclipse.  The  quantity  of  the  eclipse 
will  result  from  the  proportion 

SN  :  MN  : :  12  :  number  of  digits  eclipsed. 

Draw  from  the  centre  C  to  the  place  of  commencement^,  the 
line  C/;  and  through  the  same  point  C  raise  a  perpendicular  to 
the  ecliptic.  With  the  longitude  of  the  sun  at  the  time  of  the  be- 
ginning, calculate  its  angle  of  position  by  Problem  XIII,  and  lay  it 
off  in  the  figure,  placing  the  circle  of  declination  CP  to  the  left  if 
the  tangent  of  the  angle  of  position  be  positive,  to  the  right  if  it  be 
negative. 

Compute  also  for  the  time  of  beginning  the  angle  of  the  vertical 
circle  of  the  sun  with  the  circle  of  declination,  that  is,  the  angle 
PSZ  in  Fig.  24,  p.  47,  for  which  we  have  in  the  triangle  PSZ 
the  side  PS  =  co-declination,  the  side  PZ  —  co-latitude,  and  the 
included  angle  ZPS.  (The  requisite  formulae  are  given  in  the  Ap- 
pendix.) Form  this  angle  in  the  figure  at  the  point  C,  placing  CZ 
to  the  right  or  left  of  CP,  according  as  the  time  is  in  the  forenoon 
or  afternoon ;  CZ  will  be  the  vertical,  and  Z  the  vertex,  or  highest 
point  of  the  sun.  The  arc  Zt  on  the  limb  of  the  sun  will  be  the 
angular  distance  from  the  vertex  of  the  point  on  the  limb  at  which 
the  eclipse  commences. 

For  the  True  Times  of  Beginning,  Greatest  Obscuration,  and  End. 
The  approximate  times  of  beginning,  greatest  obscuration,  and 
end  of  the  eclipse,  being  calculated  by  the  rules  which  have  been 
given,  find  from  the  tables,  or  from  the  Nautical  Almanac,  (see 
Problem  XXXI,)  the  moon's  longitude,  latitude,  equatorial  paral- 
lax, semi-diameter,  and  hourly  motions  in  longitude  and  latitude,  for 
the  approximate  time  of  greatest  obscuration.*  With  the  moon's 
longitude  and  latitude,  and  hourly  motions  in  longitude  and  latitude, 
found  for  this  time,  calculate  the  longitude  and  latitude  for  the  ap- 
proximate times  of  beginning  and  end.  The  parallax  and  semi- 
diameter  may,  without  material  error,  be  considered  the  same 
during  the  eclipse.  With  the  moon's  true  longitude,  latitude,  and 
semi-diameter  at  the  approximate  times  of  beginning,  greatest  ob- 
scuration, and  end,  calculate  its  apparent  longitude  and  latitude, 

*  It  will,  in  general,  suffice  to  calculate  the  moon's  longitude  and  latitude  from 
the  elements  already  found  for  the  approximate  time  of  full  moon,  if  these  have 
been  accurately  determined  The  equatorial  parallax  and  semi-diameter  may  be 
found  by  interpolation  from  the  Nautical  Almanac. 


TO  CALCULATE  A  SCLAR  ECLIPSE.  327 

and  augmented  semi-diameter,  for  these  several  times,  (making  use 
of  the  relative  parallax.)  With  the  sun's  longitude  and  hourly  mo- 
tion previously  found  for  the  approximate  time  of  new  moon,  find 
his  longitude  at  the  approximate  times  of  beginning,  greatest  ob- 
scuration, and  end.  The  sun's  semi-diameter  found  for  the  ap- 
proximate time  of  new  moon,  will  serve  also  for  any  time  during 
the  eclipse.  With  the  data  thus  obtained,  calculate  by  the  formu- 
lae given  on  page  321  the  apparent  distance  of  the  centres  of  the 
sun  and  moon  at  the  approximate  times  of  the  three  phases. 

Note.  When  very  great  accuracy  is  required,  the  moon's  longi- 
tude, latitude,  equatorial  parallax,  semi-diameter,  and  hourly  mo- 
tions in  longitude  and  latitude,  must  be  calculated  directly  from 
the  tables,  or  from  the  Nautical  Almanac,  for  the  approximate 
times  of  the  beginning  and  end,  as  well  as  for  that  of  the  greatest 
obscuration. 

For  the  Beginning. 

Subtract  the  apparent  longitude  of  the  moon  at  the  approximate 
time  of  beginning  from  the  true  longitude  of  the  sun  at  the  same 
time,  and  denote  the  difference  by  a.  Do  the  same  for  the  approx- 
imate time  of  greatest  obscuration.  Subtract  the  latter  result  from 
the  former,  paying  attention  to  the  signs,  and  call  the  remainder  /c. 
Next,  take  the  difference  between  the  apparent  latitudes  of  the 
moon  at  the  approximate  times  of  beginning  and  greatest  obscura- 
tion, if  they  are  of  the  same  name  ;  their  sum,  if  they  are  of  oppo- 
site names  ;  and  denote  the  difference  or  sum,  as  the  case  may  be, 
by  n.  This  done,  compute  the  correction  to  be  applied  to  the  ap- 
proximate time  of  beginning  by  means  of  the  following  formulae  : 
log.  b  =  log.  a  4"  log.  k  +  ar.  co.  log.  n  —  10  ; 

c=\'  -b,S  =  d  +  5  —  5"; 
log  t  =  log.  (S  +  A)  +  log.  (S  -  A)  +  ar.  co.  log.  n  +  ar. 

co.  log.  c  4-  log.  L  -f  1.47712  —  20  : 
in  which 

t  =  Correction  of  approx.  time  of  beginn.  (required) ; 

a  =  Diff.  of  appar.  long,  of  Moon  and  Sun  at  approx.  time ; 

L=  Half  duration  of  eclipse  in  minutes  (known  approximately) ; 

k  =  Appar.  relative  motion  of  Sun  and  Moon  in  long,  in  the  in- 
terval L ; 

n  —  Moon's  appar.  motion  in  lat.  in  same  interval ; 

X'=  Moon's  appar.  lat. ; 

d  =  Augmented  semi-diameter  of  the  Moon ; 

6  =  Semi-diam.  of  Sun ; 

A  =  Appar.  distance  of  centres  of  Sun  and  Moon. 

b  and  c  are  auxiliary  quantities. 

First  find  the  value  of  b  by  the  first  equation,  and  substitute  it  in 
the  second.  Then  derive  the  values  of  c  and  S  from  the  second 


328  ASTRONOMICAL  PROBLEMS. 

and  third  equations,  and  substitute  them  in  the  fourth,  and  it  will 
make  known  the  value  of  t,  which  is  to  be  applied  to  the  approxi- 
mate time  of  the  beginning  of  the  eclipse  according  to  its  sign. 

The  quantities  a,  k,  n,  &c.,  are  all  to  be  expressed  in  seconds. 
The  apparent  latitude  X'  must  be  affected  with  the  negative  sign 
when  it  is  south.  The  motion  in  latitude,  n,  must  also  have  the 
negative  sign  in  case  the  moon  is  apparently  receding  from  the 
north  pole,  a  and  k  are  always  positive.* 

The  result  may  be  verified,  and  corrected,  by  computing  the  ap- 
parent distance  of  the  centres  at  the  time  found,  and  comparing  it 
with  the  sum  of  the  semi-diameters  minus  5". 

Note.  When  great  precision  is  desired,  the  quantities  k  and  n 
must  be  found  for  some  shorter  interval  than  the  half  duration  of 
the  eclipse.  Let  some  instant  be  fixed  upon,  some  five  or  ten 
minutes  before  or  after  the  approximate  time  of  the  beginning  of 
the  eclipse,  according  as  the  contact  takes  place  before  or  after. 
For  this  time  deduce  the  longitude  and  latitude  of  the  moon,  from 
the  longitude  and  latitude  at  the  approximate  time  of  beginning, 
by  means  of  their  hourly  variations  ;  and  thence  calculate  the  ap- 
parent longitude  and  latitude,  and  the  augmented  semi-diameter. 
Find  the  longitude  of  the  sun  for  the  time  in  question,  from  its 
longitude  and  hourly  motion  already  known  for  the  approximate 
time  of  beginning.  Then  proceed  according  to  the  rule  given 
above,  only  using  the  quantities  thus  found  for  the  time  assumed, 
in  place  of  the  corresponding  quantities  answering  to  the  approxi- 
mate time  of  greatest  obscuration.  L  will  always  represent  the 
interval  for  which  k  and  n  are  determined. 

For  the  End. 

Subtract  the  longitude  of  the  sun  at  the  approximate  time  of  the 
end  from  the  apparent  longitude  of  the  moon  at  the  same  time, 
Do  the  same  for  the  approximate  time  of  greatest  obscuration. 
Then  proceed  according  to  the  rule  for  the  beginning,  only  substi- 
tuting everywhere  the  approximate  time  of  the  end  for  the  approx- 
imate time  of  the  beginning,  and  taking  in  place  of  the  formula 
c  =  X'  —  ft,  the  following  : 


*  It  will  be  somewhat  more  accurate  to  use  in  place  of  k  and  n,  as  above  de- 

.  £  £'  _  k         k  k'  _  k 

fined,  the  values  of  the  following  expressions  :  -^  --  2£  —  —  —  or  —  —  3£  —  —  —  , 

^L  --  2*  H  ~n  or  -^  --  3*  "  ~H.    The  first  of  each  of  these  pairs  of  expressions 
6  3o  6  36 

is  to  be  used  in  case  the  true  time  of  beginning  is  after  the  approximate  time  ;  — 
the  second  in  the  other  case,  k'  and  n'  are  the  apparent  relative  motions  in  longi- 
tude and  latitude  during  the  last  half  of  L.  *  In  case  these  expressions  are  used 
the  following  constant  logarithm  is  to  be  employed  instead  of  that  above  given, 
viz.  0.69897. 

In  the  calculation  of  the  end  of  the  eclipse,  k  and  n  will  answer  to  the  last  half 
of  L,  and  k1  and  n'  to  the  first  half. 


TO  CALCULATE  A  SOLAR  ECLIPSE.  329 


For  the  Greatest  Obscuration. 

Take  the  sum  of  the  distances  of  the  moon  from  the  sun  in  ap- 
parent longitude  at  the  approximate  times  of  the  beginning  and  end 
of  the  eclipse,  and  call  it  k.  Take  the  difference  of  the  apparent 
latitudes  of  the  moon  at  the  same  times,  if  the  two  are  of  the  same 
name  ;  but  if  they  are  of  different  names,  take  their  sum.  Denote 
the  difference  or  sum  by  n.  Let  a1  =  the  distance  of  the  moon 
from  the  sun  in  apparent  longitude  at  the  true  time  of  greatest  ob- 
scuration ;  X'  —  the  apparent  latitude  of  the  moon  at  the  approxi- 
mate time  of  greatest  obscuration. 

k  :  n  :  :  X'  :  a'. 

Find  the  value  of  a'  by  this  proportion,  affecting  X',  n,  k,  always 
with  the  positive  sign. 

Ascertain  whether  the  greatest  obscuration  has  place  before  or 
after  the  apparent  conjunction,  by  observing  whether  the  apparent 
latitude  of  the  moon  is  increasing  or  decreasing  about  this  time ; 
the  rule  being,  that  when  it  is  increasing,  the  greatest  obscuration 
will  occur  before  apparent  conjunction ;  when  it  is  decreasing, 
after.  If  the  approximate  and  true  times  of  greatest  obscuration 
are  both  before  or  both  after  apparent  conjunction,  from  the  value 
found  for  a'  subtract  the  distance  of  the  moon  from  the  sun  in  ap- 
parejit  longitude  at  the  approximate  time  ;  but  if  one  of  the  times 
is  before  and  the  other  after  apparent  conjunction,  take  the  sum  of 
the  same  quantities.  Denote  the  difference  or  sum  by  m.  Also, 
let  D  =  duration  of  eclipse,  and  t  =  correction  to  be  applied  to  the 
approximate  time  of  greatest  obscuration.  Then  to  find  t,  we  have 
the  proportion 

k  :  m  :  :  D  :  t. 

If  the  apparent  latitude  of  the  moon  is  decreasing,  t  is  to  be 
applied  according  to  the  sign  of  m  ;  but  if  the  apparent  latitude  is 
increasing,  it  is  to  be  applied  according  to  the  opposite  sign. 

A  still  more  exact  result  may  be  had  by  repeating  the  foregoing 
calculations,  making  use  now  of  the  apparent  latitude  at  the  time 
just  found.  When  the  greatest  accuracy  is  required,  the  values  of 
k  and  n  may  be  found  more  exactly  after  the  same  manner  as  for 
the  beginning  or  end. 

For  the  Quantity  of  the  Eclipse. 

Find  by  interpolation  the  apparent  latitude  of  the  moon  at  the 
true  time  of  greatest  obscuration.  With  this,  and  the  distance  in 
longitude  a'  obtained  by  the  proportion  above  given,  compute  by 
the  formulae  on  page  321,  the  apparent  distance  of  the  centres  of 
the  sun  and  moon  at  the  time  of  greatest  obscuration.  Subtract 
this  distance  from  the  sum  of  the  apparent  semi-diameters  of  the 

42 


330  ASTRONOMICAL  PROBLEMS. 

two  bodies,  diminished  by  5",  and  denote  the  remainder  by  R 
Then, 

Sun's  semi-diam.  (diminished  by  3")  :  R  :  :  6  digits  :  number  of 
digits  eclipsed. 

When  the  apparent  distance  of  the  centres  of  the  sun  and  moon 
at  the  time  of  greatest  obscuration  is  less  than  the  difference  be- 
tween the  sun's  semi-diameter  and  the  augmented  semi-diameter 
of  the  moon,  the  eclipse  is  either  annular  or  total  ;  annular,  when 
the  sun's  semi-diameter  is  the  greater  of  the  two  ;  total,  when  it 
is  the  less. 

For  the  Beginning  and  End  of  the  Annular  or  Total  Eclipse. 

The  times  of  the  beginning  and  end  of  the  annular  or  total 
eclipse  may  be  found  as  follows  :  the  greatest  obscuration  will  take 
place  very  nearly  at  the  middle  of  the  eclipse  in  question,  and  will 
not  differ,  at  most,  more  than  five  or  eight  minutes  (according  as 
the  eclipse  is  total  or  annular)  from  the  beginning  and  end  :  to 
obtain  the  half  duration  of  the  eclipse,  and  thence  the  times  of  the 
beginning  and  end,  we  have  the  formulae 

log.  tang  6  —  log.  V  +ar.  co.  log.  a,  log.  k'=\og.  k  -f-  ar.  co.  log.  sin  0  ; 
S=$-d-  1",  orS=d—  <*  +  !"; 


loo      -      -  .  (S-A) 

log.  c  -  2  > 

log.  t  =  ar.  co.  log.  k'  +  log.  c  +  log.  D  +  1  .77815  —  10  ; 

Time  of  Begin.  =  M  —  t,  Time  of  End  =  M  +  1  : 
in  which 

M  =  Time  of  greatest  obscuration  ; 

X'  =  Moon's  apparent  latitude  at  that  time  ; 

a  =  Distance  of  moon  from  sun  in  appar.  long.  ; 

k  =  Variation  of  this  distance  during  the  whole  eclipse,  or  rela- 

tive mot.  in  appar.  long,  during  this  interval  ; 
k'  =  Moon's  appar.  mot.  on  relative  orbit  for  same  interval  ; 
&    =  Inclination  of  relative  orbit  ; 
8    =  Semi-diameter  of  sun  ; 
d  =  Augm.  semi-diam.  of  moon  ; 
A  =  Appar.  distance  of  centres  ; 

D  =  Duration  of  eclipse,  (partial  and  annular  or  total  ;) 
t    =  Half  duration  of  annular  or  total  eclipse. 

The  first  value  of  S  is  used  when  the  eclipse  is  annular,  the 
second  when  it  is  total.  The  quantities  may  all  be  regarded  as 
positive.  The  results  may  be  verified  and  corrected  by  finding 
directly  the  apparent  distance  of  the  centres  for  the  times  obtained; 
and  comparing  it  with  the  value  of  S. 


TO  CALCULATE  A  SOLAR  ECLIPSE.  331 


For  the  Point  of  the  Surfs  Limb  at  which  the  Eclipse  commences. 

Find  the  angle  of  position  of  the  sun,  and  the  angle  between  its 
vertical  circle  and  circle  of  declination,  at  the  beginning  of  the 
eclipse,  as  explained  at  page  326.  Let  the  former  be  denoted  by 
p,  and  the  latter  by  v.  Give  to  each  the  negative  sign,  if  laid  off 
towards  the  right  ;  the  positive  sign  if  laid  off  towards  the  left. 
Let  a  =•  distance  of  the  moon  from  the  sun  in  apparent  longitude 
at  the  beginning  of  the  eclipse  ;  X'  =  the  moon's  apparent  latitude 
at  the  same  time  ;  and  &  =  angular  distance  of  the  point  of  contact 
from  the  ecliptic.  Compute  the  angle  6  by  the  formula 

log.  tang  6  =  log.  X'  +  ar.  co.  log.  a  ; 

taking  it  always  less  than  90°,  and  positive  or  negative  according 
to  the  sign  of  its  tangent.  X7  is  negative  when  south  ;  a  is  always 
positive. 

Let  A  =  distance  on  the  limb  of  the  point  of  contact  from  the 
vertex.  The  above  operations  being  performed,  the  value  of  A 
results  from  the  equation 


p,  t;,  and  6  being  taken  with  their  signs. 

If  the  result  is  affected  with  the  positive  sign,  the  point  first 
touched  will  lie  to  the  right  of  the  vertex.  If  with  the  negative 
sign,  it  will  lie  to  the  left  of  the  vertex. 

Note.  The  circumstances  of  an  occultation  of  a  fixed  star  by 
the  moon  may  be  calculated  in  nearly  the  same  manner  as  those 
of  a  solar  eclipse.  The  star  in  the  occultation  holds  the  place  of 
the  sun  in  the  eclipse.  The  immersion  and  emersion  of  the  star 
correspond  to  the  beginning  and  end  of  the  eclipse.  The  elements 
which  ascertain  the  relative  apparent  place  and  motion  of  the  moon 
and  star,  take  the  place  of  those  which  ascertain  the  relative  appa- 
rent place  and  motion  of  the  moon  and  sun.  Thus  the  star's  lon- 
gitude, corrected  for  aberration  and  nutation,  (see  Problem  XXIII,) 
must  be  used  instead  of  the  sun's  longitudes  ;  the  apparent  dis- 
tances of  the  moon  from  the  star  in  latitude,  instead  of  the  moon's 
apparent  latitudes  ;  and  the  moon's  augmented  semi-diameter,  in- 
stead of  the  sum  of  the  semi-diameters  of  the  sun  and  moon.  The 
difference  of  the  longitudes,  and  the  relative  motion  in  longitude, 
must  also  now  be  reduced  to  a  parallel  to  the  ecliptic  passing 
through  the  star,  (see  Art.  490,  page  183.)  If  X  =  apparent  lati- 
tude of  star,  a  =  diff.  of  appar.  longitudes  of  moon  and  star,  and 
k  =  relative  motion  in  longitude,  we  must  substitute  in  the  formu- 
lae for  the  eclipse,  for  X',X'  —  X  ;  for  a,  a  cos  X  ;  and  for  k,  k  cos  X. 
n  will  stand  for  the  relative  motion  in  latitude,  or  for  the  variation 
of  X'  —  X. 

Example.  Required  to  calculate  an  eclipse  of  the  sun,  for  the 


332 


ASTRONOMICAL  PROBLEMS. 


latitude  and  meridian  of  New  York,  that  will  occur  on  the  18th  o1 
September,  1838. 

For  the  Approximate  Times  of  the  Phases. 

Approximate  time  of  New  Moon. 

Sept.  18d-  8h-  49m- 


175°  27'  SI"  A 

Do.  hourly  motion, 

2  26  .7 

Do.  semi-diameter, 

15  57  .0 

Moon's  longitude, 

.     175  29  19 

Do.  latitude,        .... 

47  47 

Do.  equatorial  parallax, 

53  53 

Do.  semi-diameter, 

14  41 

Do.  hor.  mot.  in  long. 

29  29 

Do.  hor.  mot.  in  lat.     . 

2  41 

Do.  appar.  long.  (Prob.  XVII),    . 

.     175   10  26 

Do.  appar.  lat.  (X7), 

2  25  N. 

Do.  augm.  semi-diameter,    . 

14  47 

Diff.  of  appar.  long,  (a), 

17     5 

Appar.  dist.  of  cen.  (A), 

17  15 

Sum  of  semi-diameters, 

30  44 

7h.  4gm. 


Sun's  longitude,  . 
Moon's  appar.  long.     . 
Do.  appar.  lat.  (X') 
Do.  augm.  semi-diameter, 
Diff.  of  appar.  long,  (a), 
Appar.  dist.  of  cen.  (A), 
Sum  of  semi-diameters, 


9h.  49t 


Sun's  longitude,  . 
Moon's  appar.  long.     . 
Do.  appar.  lat.  (X'), 
Do.  augm.  semi-diameter, 
Oiif.  of  appar.  long.  («), 
A.ppar.  dist.  of  cen.  (A), 
Sum  of  semi-diameters, 


175°  25' 
174  47 

8 


4" 
3 

12  N. 


14  49 
38  1 
38  53 
30  46 


175° 
175 


29'  58" 
36  15 

2  18  S. 
14  44 

6  17 

6  42 
30  41 


7h.  ^  *. 

8    4v 
9    49 
1049 

a 

diff.  or  k. 

X' 

diff.  or  «. 

A 

diff. 

sum  semi-d. 

2281" 
1025 
377 
1925 

1256". 
1402 
1548 

492"  N 
145   N 
138    S 
357    S 

347" 
283 
219 

2333" 
10(35 
402 
1958 

1298" 
1556 

1846" 
1844 
1841 
1839 

TO  CALCULATE  A  SOLAR  ECLIPSE.  333 

For  the  Approximate  Time  of  Beginning. 

h  =  1298",  d  =  2333"  —  1846"  =  487"  ; 

1298"  :  487" :  :  60m-  :  *  =  22m-.5 

7h.  49111. 

22 


1st  Approxi.  8h-  llm- 

7h.  49™.      >       a  =  2281"       .  X'=492"N. 

Corrections  for  22m-  447         .  133  (See  Note,  p>  324) 

8h-  llm-      .       a  =  1834         .  X'=359   N. 

a  =  1834"  ar.  co.  log.  6.73660  .         .         log.  3.26340 
V  =    359       .       log.  2.55509 

6    =  11°  4'  30"  .  tan.   9.29169          ar.  co.  cos.  0.00817 


Appar.  dist.  of  cen.       A  =  1869"    .         .     log.  3.27157 
Sum  of  semi-diam.  1846 


487"  :     23"  :  :  22m-  :  t  —  lm-  2s- 
8h.  llm. 

+  1 


2d  Approxi.  8h-  12m- 
For  the  Approximate  Time  of  the  End. 

h  =  1556",  d  =  1958"  -  1839"  =  119". 

1556"  :  119" :  :  60m-  :  t  =  4m-.6. 
10h. 

-5 


1st  Approxi.  10h-  44m- 

10h-  49m-      .      a  =  1925"       .         .         .     V  =  357/;  S. 
Corrections  for  5m>     132  17 


10h.  44m.     a  =  1793    ^    ^    .  V  =  340  S. 

a  =  1793"  .   ar.  co.  log.  6.74642    .    log.  3.25358 
X'=  340     .    .  log.  2.53148 

d  =    ...  tan.  9.27790  .  ar.  co.  cos.  0.00767 


Appar.  dist.  of  cen.  A  =  1825"       .         3.26125 
1839 

133":   14"  :  :  5m-  :  *  =  Om-.5 


334 


ASTRONOMICAL   PROBLEMS. 
10h.  44m. 

0  .5 


2dApproxi.  10h- 44m-.5 

For  the  Approximate  Time  of  Greatest  Obscuration. 

Approx.  time  of  begin.     .       8h*  12m- 
Approx.  time  of  end,         .     10    44 

2  )  18     56 

IstApproxi.     .     9     28 
For  the  True  Times  of  the  Phases. 


Approx.  time  of 


Approx.  time  of 
Greatest  Obscur. 


Approx.  time  of 
End. 


pprox 
Beginning. 

gh.  i2m-  9h-  28m<  10h>  44m- 

Sun's  longitude,  175°  26'    1".0  175°  29'    6". 8  175°32' 12".6 

Do.  semi-diam.,         15  57  .0  15  57  .0  15  57  .0 

Moon's  app.lon.  174  55  36  .7  175  27     7  .7  176     2  17  .2 

Do.  app.  lat.                5  45  .3N.  0  43  .58.           5  32  .4  S 

Do.augm.semid.        14  48  .0  14  45  .1  14  41   .7 


1856".7    1840".0 
1835  .0    1833  .7 


For  the  True  Time  of  Beginning. 


gh.  12m. 

9  28 
10  44 

a 

k 

V    I 

n 

1824".3 
119  .1 
1804  .6 

1705".2 
1923  .7 

345".3  N!  r 
43  .58,* 
332  .4  S  j 

88".8 
88  .9 

a 
A 
n 

b  = 
V 

A 

.  1824".3  . 
.  1705  .2  ... 

.   388  .8  ... 

-  8001  .1  ... 
.   345  .3 

.  log.  3.26109 
.  log.  3.23178 
ar.  co.  log.  7.41028— 

.  log.  3.90315  — 

;  =  8346  .4   ... 
.  3696  .7   ... 

ar.  co.  log.  6.07850 
.  loff.  3.56781 

V 
8 
S  -  A  .  -16  .7  . 

n  .     . 

L  76m. 


Corr.  of  approx.  time,          +  43s- .4 


.  log.  1.22272- 
ar.  co.  log.  7.41028— 

.  log.  1.88081 
Const,  log.  1.47712 

.     log.  1.63724 -f 


TO  CALCULATE  A  SOLAR  ECLIPSE. 

Corr.  of  approx.  time,  +  43s- .4 

Approx.  time,         .      8h-  12m-  0  .0 

True  time  of  begin.     8    12   43  .4,  in  Greenwich  time. 
Diff.  ofmerid.  4    56      4 


True  time  of  begin.  3    16    39  .4,  in  New  York  time. 

For  the  True  Time  of  End. 

a       .     .     1804" .6  ....  log.  3.25638 

k       .     .     1923  .7  ....  log.  3.28414 

n       .     .       288.9  .         .         ar.  co.  log.  7.53925— 

b  =      -  12016  .3  .         .         .         .  log.  4.07977— 

X'  —  332  .4  


X'  +b=c=  -12348  .7         .         .         ar.  co.  log.  5.90838— 
S-fA     .     .       3668.7         .         .         .-        .  log.  3.56451 
S-A     .     ,        —1.3         .         .         .         .  log.  0.11394— 

n ar.  co.  log.  7.53925  — 

L    .     .         .     76m log.  1.88081 

Const,  log.  1.47712 

Corr.  of  approx.  time,  —  39- 0         .  log.  0.48401  — 

Approx.  time,          .     10b-  44m-  0  .0 

True  time  of  end,  .     10     43   57  .0,  in  Greenwich  time. 
Diff.  of  merid.  4     56      4 


True  time  of  end,  .       5     47   53,       in  New  York  time. 
For  the  True  Time  of  Greatest  Obscuration. 

True  time  of  beginning,     .         .  8h-  12m-438-.4 

Do.  of  end,      .         .         .         .     10     43    57    .0 


2)  18     56    40    .4 

2d  Approx.     9     28    20    .2 

9h-  49m-     .         .     X'  =  138"     S. 
9     28  X'  =   43  .5  S. 


Diff.    21  Diff.  94  .5 


21m-  :  20s-  :  :  94".5  :  1".5 
43  .5 

•  .        .'       X'=45.  0 


336  ASTRONOMICAL  PROBLEMS. 

1705".2  388".8 

1923  .7  288  .9 


k  =  3628  .9        :  n  =  677  .7  :  :  X'  =  45".0  :  a1  =  8".4 

Time  of  beginn.  8h-  12m-  43s-  .4,  at  9h-  28m-  a  =  119".  I 
Time  of  end,     10    43     57  .0  a'=     8  .4 

D=   2    31     13  .6  m  =  -  110  .7 


3628".9  :  110".7  :  :  2h-  31ra-  13s-  .6  :  4m-  368-  .8 

9h-28       0   .0 


True  time  (nearly)  9   32     36  .8 

21m-:  4m-  37s- :  :  94".5  :  20".8 
43  .5 

At  9h-  32m-  37s-,  V  =  64  .3 

3628".9  :  677".7  :  :  64"  .4  :  12" .0 ;  at  9h<  32ra-  37s-,  a  =   8"  .4 

a'  =12  .0 

m=   3  .6 

3628".9  :  3".6  :  :  2h-  31m-  13s-.6  :      9s- .0 
9h-  32m-  36  .8 


9   32     27  .8 

True  time  of  greatest  obscur.    .    9h-  32m-  27s-. 8,  in  Greenw.  time, 
Diflf.  of  mend.  4    56       4 


True  time  of  greatest  obscur.    .    4    36     23  .8,  in  N.  Y.  time. 
For  the  Quantity  of  the  Eclipse. 

9h- 32m- 37s-     .     X'  =  64".3 
21m-  :  9s- :  :  94".5  :  0  .6 


At  nearest  approach  of  centres,    .     X'  =  63  .7 
"  "         "       .         .     a  =  12  .0 

a     .     12".0     .      ar.  co.  log.  8.92082,    .         .     log.  1.07918 
V    ,     63  .7     .  .      1.80414 


tan.  0.72496,    .  ar.  co.  cos.  0.73253 

Shortest  distance  of  centres,    64".8    .         .    log.  1.81171 
Sum  of  semi-diameters,       1837  .0 

1772  .2 
15'  54"  :  1772".2  :  :  6  :  11.14  digits  eclipsed. 


TO  CALCULATE  A  SOLAR  ECLIPSE.  337 

For  the  Situation  of  the  Point  at  which  the  Obscuration  com- 
mences. 
8h-  12m-     .     .     a     =1824",      .         .         V  =  345".3N. 

76m.  .  438.  .  .  17Q5//  .  16     76m.  .  433.  . 


Atthebeginn.      .        a  =1808,      . 
a     .     1808       .     ar.  co.  log.  6.74280 
V    .       341.6       .      .      log.  2.53352 

&  =  10°  41'  57"     .     .       tan.  9.27632 

Obliq.eclip.(Prob.X),23°  27'  47"  .  sin.  9.60005  .  tan.  9.63753 
Sun's  longitude,          175   26     3    .  sin.  8.90093  .  cos.  9.99862- 

sin.  8.50098,     tan.  9.63615  — 

Sun's  declination,  1°  49'  0"  ;  Angle  of  pos.  23°  23'  50". 
Meantime  of  begin.  3h-  16m-  39s-,  Lat.  40°  42'  40",  Dec.  1°  49'  0" 
Equa.  of  time,  5     58  90  90 

Appar.  time,     .         3    22     37,  PZ  =49  17   20,  PS  =  88  11 
60 


4 ) 202         37 

Hour  angle  P=  50°  39' 15"         .         cos.  9.80210 
Co.  lat.     PZ  =  49  17  20  tan.  0.06526 


m  =  36°23'    0"       .         .         tan.  9.86736 
Co.  dec.  PS  =88   11     0 


w'=51   48     0         .        ar.  co.  sin.  0.10466 
m  =  36   23     0         .         .  sin.  9.77320 

P=  50   39  15         .  tan.  0.08627 

S=  42   38  10         .         .  tan.  9.96413 

Angle  of  position,  .         .         —  23°  23'  50" 

Angle  from  eclip.  (&),         .         .         —  10  41  50 
Angle  of  dec.  circle  from  vertex  (S),       42  38  10 

90 


Angular  dist.  of  point  first  touched  from  vertex,  98  32,  to  the  right 
For  the  Beginning  and  End  of  the  Annular  Eclipse. 

Approx.  time,  9h>  32m-  27S\8  =true  time  of  greatest  obscur. 

At  this  time,  a  =  12".2,\'  =63".7. 

a  =  12".2         .          ar.  co.  log.  8.91364     .         .     log.  1.08636 
V=63  .7         .         .  log.  1.80414 

6  =  79°  9'  30"  .  tan.  0.71778    .    ar.  co.  cos.  0.72564 

A  =  64".9      .         ,     log.  1.81200 
43 


ASTRONOMICAL  PROBLEMS. 


=  135".8  .log.  2.  13290,4  =79°  9;  30"  .  ar.  co.  sin.  0.00783 
*  log.  3.55977 


S  -  A=  6  .2  .  log.  0.79239,  &=3628".9 
2  )  2.92529,  k1 

1.46264 
D=152m- 


Time  of  greatest  obscur.   .  4  36    23  .8 
Formation  of  ring,     . 
Rupture  of     do. 


ar.  co.  log.  6.43240 

.  .  1.46264 
.  .log.  2.18184 
Const,  log.  1.77815 

.    log.  1.85503 


.    4  35    12  .2,  New  York  time. 

«          a 


4  37    35  .4 


PROBLEM  XXXI. 

To  find  the  Mootfs  Longitude,  Latitude,  Hourly  Motions,  Equa- 
torial Parallax,  and  Semi-diameter,  for  a  given  time,  from  the 
Nautical  Almanac. 

Reduce  the  given  time  to  mean  time  at  Greenwich  ;  then, 
For  the  Longitude. 

Take  from  the  Nautical  Almanac  the  calculated  longitudes  an- 
swering to  the  noon  and  midnight,  or  midnight  and  noon,  next  pre- 
ceding and  next  following  the  given  time.  Commencing  with  the 
longitude  answering  to  the  first  noon  or  midnight,  subtract  each 
longitude  from  the  next  following  one  :  the  three  remainders  will 
be  the  first  differences.  Also  subtract  each  first  difference  from 
the  following  for  the  second  differences,  which  will  have  the  plus 
or  minus  sign,  according  as  the  first  differences  increase  or  de- 
crease. 

Find  the  quantity  to  be  added  to  the  second  longitude  by  rea- 
son of  the  first  differences,  by  the  proportion,  1 2h* :  excess  of  given 
time  above  time  of  second  longitude  :  :  second  first  difference : 
fourth  term. 

With  the  given  time  from  noon  or  midnight  at  the  side,  take  from 
Table  XCIII  the  quantities  corresponding  to  the  minutes,  tens  of 
seconds,  and  seconds,  of  the  mean  or  half  sum  of  the  two  second 
differences,  at  the  top  :  the  sum  of  these  will  be  the  correction  for 
second  differences,  which  must  have  the  contrary  sign  to  the  mean. 

The  sum  of  the  second  longitude,  the  fourth  term,  and  the  cor 
rection  for  second  differences,  will  be  the  longitude  required. 


TO  FIND  MOON'S  LONG.,  ETC.,  FROM  NAUTICAL  ALMANAC.     339 

For  the  Latitude. 

Prefix  to  north  latitudes  the  positive  sign,  but  to  south  latitudes 
the  negative  sign,  and  proceed  according  to  the  rules  for  the  lon- 
gitude, only  that  attention  must  now  be  paid  to  the  signs  of  the  first 
differences,  which  may  either  be  plus  or  minus. 

The  sign  of  the  resulting  latitude  will  ascertain  whether  it  is 
north  or  south. 

For  the  Hourly  Motion  in  Longitude. 

Solve  the  proportion,  1 2h-  :  given  time  from  noon  or  midnight 
:  :  half  sum  of  second  differences :  a  fourth  term ;  which  must  have 
the  same  sign  as  the  half  sum  of  the  second  differences. 

Take  the  sum  of  the  second  first  difference,  half  the  mean  of 
the  second  differences,  with  its  sign  changed,  and  this  fourth  term, 
and  divide  it  by  12 :  the  quotient  will  be  the  required  hourly  mo- 
tion in  longitude. 

For  the  Hourly  Motion  in  Latitude. 

With  the  given  time  from  noon  or  midnight,  the  second  first 
difference  of  latitude,  and  the  mean  of  the  second  differences,  find 
the  hourly  motion  in  latitude  in  the  same  manner  as  directed  for 
finding  the  hourly  motion  in  longitude.  When  the  hourly  motion 
is  positive,  the  moon  is  tending  north ;  and  when  it  is  negative, 
she  is  tending  south. 

For  the  Semi-diameter  and  Equatorial  Parallax. 

The  moon's  semi-diameter  and  equatorial  parallax  may  be  taken 
from  the  Nautical  Almanac,  with  sufficient  accuracy,  by  simple 
proportion,  the  correction  for  second  differences  being  too  small  to 
be  taken  into  account,  unless  great  precision  is  required. 

Corrections  for  Third  and  Fourth  Differences. 

When  the  moon's  longitude  and  latitude  are  required  with  great 
precision,  corrections  must  also  be  applied  for  the  third  and  fourth 
differences.  To  determine  these,  take  from  the  Almanac  the  three 
longitudes  or  latitudes  immediately  preceding  the  given  time,  and 
the  three  immediately  following  it,  and  find  the  first,  second,  third, 
and  fourth  differences,  subtracting  always  each  number  from  the 
following  one,  and  paying  attention  to  the  signs.  With  the  given 
time  from  noon  or  midnight  at  the  side,  and  the  middle  third 
difference  at  the  top,  take  from  Table  XCIV  the  correction  for 
third  differences,  which  must  have  the  same  sign  as  the  middle 
third  difference  when  the  given  time  from  noon  or  midnight  is  less 
than  6  hours  ;  the  contrary  sign,  when  the  given  time  is  more  than 
6  hours. 


840  ASTRONOMICAL  PROBLEMS. 

With  the  given  time,  and  half  sum  of  fourth  differences,  take 
from  Table  XCV  the  correction  for  fourth  differences,  giving  it 
always  the  same  sign  as  the  half  sum. 

The  sum  of  the  third  longitude  or  latitude,  the  proportional  part 
of  the  middle  first  difference  answering  to  the  given  time  from 
noon  or  midnight,  and  the  corrections  for  second,  third,  and  fourth 
differences,  having  regard  to  the  signs  of  all  the  quantities,  will  be 
the  longitude  or  latitude  required. 


APPENDIX. 


TRIGONOMETRICAL   FORMULAE.* 


I.  RELATIVE  TO  A  SINGLE  ARC  OR  ANGLE  <z. 

1.  sin2  a  +  cos2  a  =  I 

2.  sin  a  =  tan  a  cos  a 

tana 

3.  sm  a  =  — • ; 


cos  a  — 


I  +  tan2  a 

sin  a 

5.  tan  a  = 

cos  a 

I          cos  a 

6.  cot  a  = =  — 

tan  a       sm  a 

7.  sin  a  =  2  sin  |  a  cos  | 

8.  cos  a  =  1  —  2  sin2  J  a 

9.  cos  a  =  2  cos2  £  a  —  1 


10.  tani 

11.  cot£ 

.  . 

12.  tan2  i  a  =  —  ; 

1  + 


COS 

sin  a 


1  —  cos  a 
1  —  cos  a 


cos  a 

13.  sin  2  a  =  2  sin  a  cos  a 

14.  cos  2  a  =  2  cos2  a  —  1  =  1  —  2  sin*  a 

II.  RELATIVE  TO  Two  ARCS  a  AND  6,  op  WHICH  a  is  SUPPOSED 

TO  BE  THE  GREATER. 

15.  sin  (a  +  b)  —  sin  a  cos  6  +  sin  b  cos  a 

16.  sin  (a  —  b)—  sin  a  cos  b  —  sin  6  cos  a 

17.  cos  (a  +  b)  =  cos  a  cos  b  —  sin  a  sin  b 

9  The  radius  is  supposed  to  be  equal  to  unity  in  all  of  the  formulae. 


APPENDIX. 


16.     cos  (a  —  b)  •=•  cos  a  cos  b  +  sin  a  sin 
tan  a  -\-  tan  6 

1Q           f-,^    ln   _L  M   

6 

(«-*)• 
•(«  +  *) 

(a  -6) 
(a  -6) 

(a  +  6) 

ly.        Idll  IW  T  t/J          ,                                  7 

1  —  tan  a  tan  6 
,x      tan  a  —  tan  6 

OA         tan  1/7          n  1  — 

<6U.       Idll  It*          Wl 

l+tanatano 
21  .     sin  a  -h  sin  6  =  2  sin  |  (a  +  b)  cos  £ 
22.     sin  a  —  sin  6  =  2  sin  |  (a  —  &)  cos  J 
23.     cos  a  -|-cos&  =  2  cos  £  (a  +  &)  cos  £ 
24.     cos  b  —cosa  =  2  sin  \  (a  +  b)  sin  | 
,       sin  (a  +  b) 

cos  a  cos  b 
,       sin  (a  —  b) 

9fi        tin  rr         tin  n  — 

cos  a  cos  b 
"7       rota   ^rot/>-Sin(fl  +  6) 

sm  a  sm  o 

sin  a  sin  6 
on      sin  a  +  sin  6      tan  %  (a  +  6) 

sin  a  —  sin  6      tan  |  (a  —  6) 
cos  b  +  cos  a      cot  |(a  -f  6) 

cos  b  —  cos  a      tan  £  (a  —  6) 
tan  a  -f  tan  6      cot  b  +  cot  a      sin 

tan  a  —  tan  b      cotb  —  cot  a      sin  (a  —  6) 
oo      cot  b  —  tan^z      cot  a  —  tan  b      cos  (a  +  6) 

cot  b  -|-  tan  a      cot  a  +  tan  b      cos 
33.     sin2  a  —  sin2  6  =  sin  (a  +  6)  sin  (a  - 
34.     cos2  a  —  sin2  b  =  cos  (a  +  6)  cos  (a  • 
35.     1  ±  sin  a  =  2  sin2  (45°  ±  i  a) 

(«-*) 

-6) 

-6) 

1  =F  sin  a 

37.    L±.?L?  =  tan  (45°  ±  i  a) 
cosa 

00      1  —  sin  a      sin2  (45°  —  i  a) 

38.             -                                                                      .     a    - 

1  —  cos  a              sin2  £  a 
a        1  +  sin  b  _  sin2  (45°  +46) 

1  +  cosa            cos2  ^  fl 

1  —  tan  b  _        .            .  . 

TRIGONOMETRICAL   FORMULAE.  343 

42.  sin  a  cos  b  =  ±  sin  (a  -\-b)  +  I  sin  (a  —  b) 

43.  cos  a  sin  b  =  1  sin  (a  +  b)  —  \  sin  (a  —  b) 

44.  sin  a  sin  b  —  I  cos  (a  —  b)  —  J  cos  (a  +  b) 

45.  cos  a  cosb—±  cos  (a  +  b)  + ±  cos  (a  —  b) 

III.  TRIGONOMETRICAL  SERIES. 


46.  ^ 

f   .                       a3 

+ 
+ 
+ 

«5 

A 

a6 

-  4-  Ar 

"2.3 
«2 

2. 

3.4.5 
a4 

_3 

2.3.4 
205 

2. 

4_    A 

3.4. 
7a7 

5.6  f 

+  &C. 

&c. 

3 

1       a 

C0i  a  ~~  a       3  " 

«• 

3.5 

1  32. 

5.7 

32. 

5 

33.5. 

7 

Let  a  —  length  of  an  arc  of  a  circle  of  which  the  radius  is  1,  and 
(a")  =  number  of  seconds  in  this  arc,  then  to  replace  an  arc  ex- 
pressed by  its  length,  by  the  number  of  seconds  contained  in  it,  we 
nave  the  formula 

47.  a  =  (a")  sin  1"  ;  log.  sin  I"  =^"6.685574867. 

IV.  DIFFERENCES  OF  TRIGONOMETRICAL  LINES. 

48.  A  sin  x  —  +  2  sin  \  A  x.  cos  (x  +  |  A  x) 

49.  A  cos  x  =  —  2  sin  £  A  x.  sin  (x  +  £  A  x) 

sin  A  x 


50.     A  tan  x  = 
51. 


cos  x.  cos  (x  +  A  x) 

sin  A  x 
sin  x.  sin  f(x  +  A  x) 

V.  RESOLUTION  OF  RIGHT-ANGLED  SPHERICAL  TRIANGLES.* 

Table  of  Solutions. 

Given.  ,  Required.  Solution. 

Hypothen.  (  side  op.  giv.  ang.   52     sin  x  =  sin  h  .  sin  a 
and      <  side  adj.  giv.  ang.   53     tan  x  —  tan  h  .  cos  a 
an  angle    (.  the  other  angle       54     cot  x  =  cos  h  .  tan  a 

cos  h 


Hypothen 

and 
a  side 


the  other  side         55     cos  x  = 


cos  s 


ang.  adj.  giv.  side  56     cos  x  =  tan  s .  cot  h 

.,  .  si 

ang.  op.  giv.  side   57    sin  x  =  — 


sin  s 


sn 


*  Baily's  Astronomical  Tables  and  Formal®. 


344  APPENDIX. 

,     the  hypothen.         58     sin  x  —  — 
A  side  and  I  sin  a 

the  angle  J  the  other  side         59     sin  x  =  tan  s  .  cot  a  ^  | 

opposite  __       .  cos  a 

the  other  angle       60     sin  x  = 

I  cos  s          )  | 

A  side  and  f  the  hypothen.  61  cot  x  =  cos  a  .  cot  s 
the  angle  <  the  other  side  62  tana:  —  tan  a  .  sin  s 
adjacent  [the  other  angle  63  cos  x  —  sin  a  .  cos  s 

{the  hypothen.         64     cos  a?  =  rectang.  cos.  of  the 
giv.  sides 
an  angle  65      cot  x  =  sin  adj.  side   x  cot. 

op.  side 

{the  hypothen.         66      cos  a?  =  rectang.  cot.  of  the 
given  angles 
cos.  opp.  ang. 
a  side  67      cos  x  =  -. £*- — 

sin.  adj. ang. 

In  these  formulae,  x  denotes  the  quantity  sought. 
a  =  the  given  angle 
*  =  the  given  side 
h  =  the  hypothenuse. 


The  formulae  for  the  resolution  of  right-angled  spherical  trian- 
gles are  all  embraced  in  two  rules  discovered  by  Lord  Napier,  and 
called  Napier1  s  Rules  for  the  Circular  Parts.  The  circular  parts, 
so  called,  are  the  two  legs  of  the  triangle,  or  sides  which  form  the 
right  angle,  the  complement  of  the  hypothenuse,  and  the  comple- 
ments of  the  acute  angles.  The  right  angle  is  omitted.  In  re- 
solving a  right-angled  spherical  triangle,  there  are  always  three  of 
the  circular  parts  under  consideration,  namely,  the  two  given  parts 
and  the  required  part.  When  the  three  parts  in  question  are  con- 
tiguous to  each  other,  the  middle  one  is  called  the  middle  part,  and 
the  others  the  adjacent  parts.  When  two  of  them  are  contiguous, 
and  the  third  is  separated  from  these  by  a  part  on  each  side,  the 
part  thus  separated  is  called  the  middle  part,  and  the  other  two  the 
opposite  parts.  The  rules  for  the  use  of  the  circular  parts  are  (the 
radius  being  taken  =  1 ), 

1 .  Sine  of  the  middle  part  =  the  rectangle  of  the  tangents  of  the 
adjacent  parts. 

2.  Sine  of  the  middle  part  =  the  rectangle  of  the  cosines  of  the 
opposite  parts. 

PARTICULAR  CASES  OF  RIGHT-ANGLED  SPHERICAL  TRIANGLES. 

Equations  52  to  67,  or  Napier's  rules,  are  sufficient  to  resolve 
all  the  cases  of  right-angled  spherical  triangles  ;  but  they  lack  pre- 
cision if  the  unknown  quantity  is  very  small  and  determined  by 


RESOLUTION  OF  SPHERICAL  TRIANGLES.  345 

means  of  its  cosine  or  cotangent  ;  or,  if  the  unknown  quantity  is 
near  90°,  and  given  by  a  sine  or  a  tangent  :  in  these  cases  the  fol- 
lowing formulae  may  be  used  : 

cos(B  +  C) 
68. 


cos  (B  —  C) 

sin  (a  —  c) 

69.  tan2  IB  =  .    ;     ,     : 

sin  (a  -f  c) 

70.  tan2  ic  =  tan  £  (a  +  b)  tan  1  (a  —  b) 

71.  tan  (45°  —  16)  =  ^  tan  (45°-  #),  tan  a?  =  sin  a  sin  B 

72.  tan2  16  -tan          -^+45°     tan 


a  is  the  hypothenuse,  B,  C,  the  acute  angles,  and  b,  c,  the  sides 
opposite  the  acute  angles. 

VI.  RESOLUTION  OF  OBLIQUE-ANGLED  SPHERICAL  TRIANGLES. 
General  Formula. 

Let  A,  B,  C,  denote  the  three  angles  of  a  spherical  triangle,  and 
a,  by  c,  the  sides  which  are  opposite  to  them  respectively, 
sin  A  _  sin  B  _  sin  C 
sin  a       sin  b      sin  c 

or,  the  sines  of  the  angles  are  proportional  to  the  sines  of  the  op* 

posite  sides. 

74.  cos  c  —  cos  a  cos  b  +  sin  a  sin  b  cos  C 

75.  cos  c  =  cos  (a  —  b)  —  2  sin  a  sin  b  sin2  £C 

76.  cos  C  =  sin  A  sin  B  cos  c  —  cos  A  cos  B 

77.  sin  a  cos  c  —  sin  c  cos  a  cos  B  +  sin  b  cos  C 

78.  sin  a  cot  c  =  cos  a  cos  B  +  sin  B  cot  C 

79.  sin  a  cos  B  =  sin  c  cos  b  —  sin  b  cos  c  cos  A 

Case  i.  Given  the  three  sides,  a,  by  c. 
To  find  one  of  the  angles. 

•  2  i  A       sin  (It  —  6)  sin  (k  —  c) 

80.  sin  iA  = : — : — : 

sin  b  sin  c 

or» 

81 .  COS2  I A  = : — ji : 

sm  b  sin  c 

82.  A  =  1±|±£.       ^Ill^pl^lj*^ 

Case  ii.  Given  the  three  angles.  A,  B,  C 

To  find  one  of  the  sides. 

—  cos  K  cos  (K  —  A) 

83.  sina£tf  = .    p    .' ' 

sm  B  sin  C 

44 


346 


APPENDIX. 


or, 


84. 


85.   K= 


sm  B  sm  C 


Case  in.  Given  two  sides  a  and  &,  and  the  included  angle  C. 
1°.  To  find  the  two  other  angles  A  and  B. 


Napier's  Analogies. 


87. 


2°.   To  find  the  third  side  c. 


88. 


or, 


tan  £c=  tan  £  (a  +  b). 


cosi(A+B) 


cos*(A~B) 
or  equa.  73. 

Case  iv.  Given  ft#o  angles  A  awe?  B,  and  the  adjacent  side  c. 
1°.  To  find  the  other  two  sides,  a  and  6. 


sin  i  (A 


Napier's  Analogies. 


90.    tan  £  (a— b)  =tai.  2^.  -.—  ,  , .  .  ^v  , 

sm  i  (A+  B)  J 

2°.  To  find  the  third  angle  C. 


91. 


*  sin  £  (a —  b) 


or 


cos  $(a+b) 
'  cos  2  (a—b) 
or  equa.  73. 

Case  y.  Given  two  sides  a,  b,  and  an  opposite  angle  A, 

To  find  the  other  opposite  angle  B ;  take  equation  73,  or  the 
proportion ;  sines  of  the  angles  are  as  sines  of  the  opposite  sides. 
(For  the  methods  of  determining  the  remaining  angle  and  side,  see 
page  348,  Case  3.) 

Case  vi.   Given  two  angles  A,  B,  and  an  opposite  side  a. 
To  find  the  other  opposite  side  b  •  sines  of  the  angle  are  proper- 


RESOLUTION  OF  SPHERICAL  TRIANGLES.  347 

tional  to  the  sines  of  the  opposite  sides.     (For  the  methods  of  de- 
termining the  remaining  side  and  angle,  see  page  348,  Case  4.) 

OTHER  METHODS  OF  RESOLVING  OBLIQUE-ANGLED  SPHERICAL 
TRIANGLES.* 

Except  when  three  sides  or  three  angles  are  given,  the  data 
always  include  an  angle  A,  and  the  adjacent  side  &,  besides  a  third 
nart.     The  required  parts  in  the  different  cases  may  be  found  by 
*e  following  formulae,  and  formula  73. 

cot  n  =  tan  A  cos  b 


98 


tan  B 


From  the  angle  C  (Fig.  124)  a  perpendicular  CD  is  let  fall  upon 
the  opposite  side  c,  which  divides  the 
triangle  into  two  right-angled  trian- 
gles, that  are  resolved  separately.  In 
the  one,  ACD,  A  and  b  are  known, 
and  it  is  easy  to  find  the  other  parts, 
which,  joined  to  the  third  given  part, 
serve  to  resolve  the  second  right-an- 
gled triangle  BCD,  and  determine  the 
unknown  quantity  required,  m,  m'  A 
denote  the  two  segments  of  the  base  ;  n,  n'  the  two  parts  of  the 
angle  C  ;  and  k  the  perpendicular  arc  CD. 

It  must  be  observed,  that  if  the  perpendicular  CD  fell  without 
the  triangle,  m  and  mr,  n  and  n1  would  have  contrary  signs  ;  this 
happens  when  the  angles  A  and  B  at  the  base  are  of  different  kinds, 
(the  one  Z_,  the  other  >90°).  When  it  is  not  known  whether  this 
circumstance  has  place  or  not,  the  problem  is  susceptible  of  two 
solutions. 

The  detail  of  the  different  cases  is  as  follows  :  the  data  are  A, 
6,  and  another  arc  or  angle. 

Case  1.  Given  two  sides  and  the  included  angle  ;  or  6,  c,  A. 

Equation  92  makes  known  m,  94  m',  which  may  be  negative, 
(what  the  calculation  shows,)  96  a,  98  B,  and  equation  73,  (page 
345,)  C,  which  is  known  in  kind.  .  * 

Case  2.   Given  two  angles  and  the  adjacent  side;  or  A,  C,  b. 

Equation  93  makes  known  n,  95  n',  which  may  be  negative, 
(what  the  calculation  shows,)  97  B,  99  a  ;  finally,  equation  73 
(page  345)  gives  c,  which  is  known  in  kind. 

»  FranecBur's  Practical  Astronomy. 


3  £8  APPENDIX. 

Case  3.   Given  two  sides  and  an  opposite  angle;  crb,  o,  A. 

Equation  92  gives  m,  96  m',  94  c,  98  and  73  B  and  C ; 

or  else,  93  gives  w,  99  ft',  95  C,  97  and  73  B  and  c. 

This  problem  admits  in  general  of  two  solutions.  In  effect,  the 
arc  m'  or  angle  n'  being  given  by  its  cos.,  may  have  either  the 
sign  -f  or  — ;  there  are  then  two  values,  for  c,  and  also  for  C.  m' 
and  n'  enter  into  equations  97  and  98  by  their  sines,  whence  result 
therefore  also  two  values  of  B. 

Case  4.   Given  two  angles,  and  an  opposite  side;  or  A,  B,  b. 

Equation  92  gives  m,  98  m',  94  c,  96  a,  and  equation  73  makes 
known  C  ; 

or  else  93  gives  n,  97  n',  95  C,  99  and  73  a  and  c. 

There  are  also  two  solutions  in  this  case  ;  for,  m1  or  n'  is  given 
by  a  sin.,  and  therefore  two  supplementary  arcs  satisfy  the  ques- 
tion. Thus  c  in  94,  and  a  in  96,  receive  two  values ;  same  for 
C  in  95,  and  a  in  99,  &c. 

Instead  of  solving  the  two  right-angled  triangles,  into  which  the 
oblique-angled  triangle  is  divided,  by  equations  92  to  99,  we  may 
employ  Napier's  rules,  from  which  these  equations  have  been  ob- 
tained. 

Isosceles  Triangles. 

When  the  triangle  is  isosceles,  B  =  C,  b  =  c,  the  perpendicular 
arc  must  be  let  fall  from  the  vertex  A,  and  the  equations  furnished 
by  Napier's  rules,  become  very  simple.  We  find 

101.  sin  £  a  =  sin  £  A  sin  b 

102.  tan  \  a  =  tan  b  cos  B 

103.  cos  b     =  cot  B  cot  £  A 

104.  cos  $  A  =  cos  £  a  sin  B 

The  knowledge  of  two  of  the  four  elements  A,  B,  #,  b,  which 
form  the  isosceles  triangle,  is  sufficient  for  the  determination  of  the 
two  others. 


INVESTIGATION  OF  ASTRONOMICAL  FORMULAE. 

Formula  for  the  Parallax  in  Right  Ascension  and  Declination, 
and  in  Longitude  and  Latitude.  (See  Article  120,  page  55.) 

Let  s  (Fig.  125)  be  the  true  place 
of  a  star  seen  from  the  centre  of  the 
>     earth,  s'  the  apparent  place,  seen  from 
a  point  on  the  surface  of  which  z  is 
the  zenith,  the  latitude  being  /.     The 
displacement  ss'  =  p  is  the  parallax 
in  altitude,  which  takes  effect  in  the  vertical  circle  zs' ;  p  is  the 


PARALLAX  IN  RIGHT  ASCENSION  AND  DECLINATION.  349 

pole  ;  the  hour  angle  zps  =  q  is  changed  into  zps',  and  sps1  =  a 
is  the  variation  of  the  hour  angle,  or  the  parallax  in  right  ascen- 
sion ;  the  polar  distance  ps  =  d  is  changed  into  ps' ;  the  differ- 
ence 8  of  these  arcs  is  the  parallax  in  declination  or  of  polar  dis- 
tance.* We  have,  (For.  73,  p.  345,) 

sin  s' :  ships  (d) :  :  sin  sps'  (a)  :  sin  ssf  (p), 

sin  zps'  (q  -|-a)  :  sin  zs'  (Z) :  :  sin  s' :  sinpz  (90°—  I). 

Multiplying,  term  by  term,  we  obtain 

sin  sr  sin  (<?  +  «):  sin  d  sin  Z  :  :  sin  a  sin  s' :  sin  p  cos  / ; 

sin  p  cosl   .    . 

whence,  sin  a  =    .    *    . — ^  sin  (q  +  a) . 

sin  c?  sin  Z 

Or,  substituting  forp  its  value  given  by  equa.  (8,)  p.  51,  and 
replacing  H  by  P, 

sin  P  cos  /  .     ,          x         ,.. 

sin  a  = : — - —  sin  (q  +  a)  .  .  .  (A). 

sin  d 

This  equation  makes  known  a  when  the  apparent  hour  angle 
zps  —  q  +  a,  seen  from  the  earth's  surface,  is  given  ;  but  if  we 
know  the  true  hour  angle  zps  —  q,  seen  from  the  centre  of  the 
earth,  developing  sin  (q  +  a)>  (For.  15,  p.  341),  and  putting 
sin  P  cos  I  _ 

sin  d 

sin  a  =  ?7i  (sin  q  cos  a  -f-  sin  a  cos  q), 
or,  dividing  by  sin  a, 

1  =  77i  (sin  q  cot  a  +  cos  q)  ; 
whence,  by  transformation, 

ffi  sin  <7  .  „   .  .  ,    v 

tan  a  = 2 —  =  m  sm  #  -f-  m*  sm  <TT  COs  g  (very  nearly.) 

1  —  m  cos  q 

Restoring  the  value  of  m, 

sin  P  cos  Z   .        .    /sinPcosZXa    . 

tan  a  = : — - —  sm  a  +  I : — ; —  I    sm  q  cos  ». 

sm  d  \      smd     / 

Putting  the  arc  a  in  place  of  its  tangent,  and  P  in  place  of  sin  P, 
and  expressing  these  arcs  in  seconds,  (For.  47,  p.  343,)  there 
results, 


Pcos/ 


.     /P  COS  /\2     .  .  _ 

a  =  — : — j-  sin  q  +  I  — : — -j~  ]    sin  q  cos  q  sm  1"  ...  (B). 
smd  V  smd  / 

The  parallax  in  declination  (8)  is  the  difference  of  the  arcs  ps 
(=  d)  and  X  (=  d  +  8.)  Let  *s  =  z,  and  zs'  =  Z.  The  trian- 
gles zjos  and  zps'  give  (For.  74  and  73), 

cos  d  —  sin  /  cos  z     cos  (<?  +  £)—  sin  /  cos  Z 

1°.  cos  pzs  = =-T = 3 —    %  .    _ , 

cos  /  sin  z  cos  I  sm  Z 

*  Francceur'a  Uranography,  p.  418. 


350  APPENDIX. 

_  sin  d  sin  q  _  sin  (d  +  #)  sin  (?  +  «) 
•^  sin  z  sin  Z 

From  the  first  equation  we  derive 

,   «      cos  d  sin  Z  —  sin  I  cos  z  sin  Z        .  ' 
cos  (a  -f-  o)  =  ---  :  -----  h  sin  /  cos  Z 

smz 

_  cos  6?  sin  Z  —  sin  /  (cos  z  sin  Z  —  sin  z  cos  Z) 

sin  z 
_  cos  d  sin  Z  —  sin  Z  sin  (Z  —  z) 

sin  z 
or,  (equ.  8,  p.  51,) 

sinZ  ,        ,        •    r»    •    7v 
=  -  —  (cos  a  —  sin  P  sin  /)  : 
sm  z  v 

from  the  second, 

sinZ  _  sin  (d  -f  5)   sin  (q  -f  a) 

sin  z  sin  d  sin  q 

substituting, 

,  _       ~       sin  (d  +  5)    sin  (7  -f  «)  /         ,        •    r»    •    A 
cos  (d  -h  <5)  =  -  ^  —  5—^  .  -  4^—  -  —  -  (cos  rf  —  sm  P  sin  /) 
sin  d  sm  q 

cos  (df  +  ^)  _  sin  (q  +  a)  /cos  d       sin  P  sin  l\ 
sin  (c?  +  ^)  sin  q       Vsin  d  sin  d     ) 

*  /^  i  *\       s^n  (7  +  a)  ^     ,  ^      sin  P  sin  ^         /r«\ 
cot  (rf  -f  5)  =  --  ^  -  -  I  cot  d  --  :  —  j—  I  ...  (C). 
sm  q       \  sin  d     / 

~  sin  P  sin  / 

Put  tan  x  =  -  :  —  -j—  ; 
sm  d 

then,  cot  (d  +  6)  =  ^^-^  (cot  d  -  tan  x) 

sm  (] 

__sin  (g  +  a)  /cosd      sin  a?\ 
sin^      Vsin  6?     cosx/ 

_  sin  (#  +  a)   cos  6?  cos  a?  —  sin  d  sin  a: 

sin  q  sin  rf  cos  a? 

sin       +  a   c 


sin  ^  sin  d  cos 


The  apparent  polar  distance  (d  +  5)  being  computed  by  either 
of  the  formulae  (C)  and  (D),  we  have  8  =  (d  +  5)  —  d. 

Formulas  may  be  obtained  that  will  give  the  parallax  in  declina  . 
tion  without  first  finding  the  apparent  declination,  (except  approx 
iinately.) 

From  equa.  (C)  we  obtain 

sin  P  sin  /  sin  q  cot  (d  +  5) 

-  :  --  j  —  ~~  COt  rf  —  -  :  —  7  -  ,  -  r  -  , 

sm  d  sm  (q  -f  «) 


PARALLAX  IN  RIGHT  ASCENSION  AND  DECLINATION.          351 

and  we  also  have 

rx      cos  d     cos  (d  +  <$)  sin  <S 

cot  d  —  cot  (d  +  <5)  =  - — -, .    ,  ,  ,     '  =  .     ,   .    , ,  ,   -v  ; 

sm  rf      sin  (rf  4-  <$)      sm  d  sm  (d  +  d) 

the  sum  of  these  equations  gives 

sin  P  sin  /  «  /.  sin  q     \  sin  d 


sin  d!  V        sin  (q  +  <*)'      sin  d  sin  (d  +  £)' 

sin  (/      __  sin  (y  -f  «)  -  sin  q 

w,  l    ~        :      ~.        .        x  —  : ~ . r 

sm  (q  +  a)  sm  (q  +  a) 

_  2  sin  |  a  cos  (q  +  j  a)  _  sin  a  cos  (y  +  j  a) 

: — "7 r —      — : 7 : . : \£  OF.   ^&/tf,    1  u I 

sin  (^  +  a)  sin  (q  -f"  a)  cos  ^  a 

cos  (7  -f  i  «)  sin  P  cos  I  ,    % 

•  = .     ,    •  -T ,  by  equa.  (A). 

sm  c?  cos  i  a 


sin  d  sin  (d  +  <*)' 
or,  sin  <5  =  sin  P  sin  Z  sin 

cos  (d  +  8)  cos  (<7+£a)  sin  P  cos 


cos    a 
=  sin  P  sin  /  [sin  (d  +  5)  —  tan  y  cos  (rf  -f-  $)], 

.  .  coil  cos  (»  -f  ^  «) 

making  tan  y  =  -  ~  ; 

cosia 

,  .    .      sin  P  sin  I   .    .  ,  .   .  ._. 

whence,          sin  a  =  •  --  sm  (d  +  5  —  y)  .  .  .  (F). 
cosy 

To  facilitate  the  calculation,  the  sines  of  5  and  P  in  eqs.  (E) 
and  (F),  may  be  replaced  bv  the  arcs. 

To  obtain  an  expression  for  the  parallax  in  declination  in  terms 
of  the  true  declination,  develope  ain  (d  +  <5  —  y)  in  equation  (F) 
which  gives 

.    ,,      sin  P  sin  /  r  .     ,  ,  ,   „ 
sm  d  —  --  [sm  (d  +  o)  cos  y  —  sin  y  cos  (d  +  ^)]  ; 

developing  sin  (d  +  <5)  and  cos  (d  +  5),  and  reducing,  we  have 

.    .     sinP 
sm  5  = 

co 

dividing  by  cos  <J, 


.    .     sinPsin/r  . 

sm  5  =  -  ,  --  [sm  (d  —  y)  cos  o  +  cos  (a  —  y)  sin 
cos  y 


~"y)  +  COS  (rf~ 


352  APPENDIX. 


cos  y 
whence  tan  6-  -- 


sin  P  sin  /    .    ,  ,      x 
sm  (d—y) 


1  ---  cos  (a  —  v) 
cosy 

sin  P  sin/   .     .,       v  ,  /sin  P  sin  A2 
=  ---  sin  (d—y)  -f  I  ---  I  x 
cosy  V     cosy     / 

sin  (d  —  y)  cos  (d—y)  (very  nearly  ;) 

or,  replacing  tan  8  and  sin  P  by  5  and  P,  expressing  these  arcs  in 
seconds,  (For.  47,  p.  343),  and  reducing  by  For.  13,  p.  341, 

PsinJ    .  x   ,  /P  sin  /\2  sin  I"   . 

J  =  -  sm  (d—  y)  +  [  --  I   -   -  —  sm2(d—  y)    .  .  .  (G.) 
cos  y  V  cos  y  /       2 

If  the  place  of  a  body  be  referred  to  the  ecliptic,  similar  formu- 
lae will  give  the  parallax  in  latitude  and  longitude,  but  as  the 
ecliptic  and  its  pole  are  continually  in  motion  by  virtue  of  the  di- 
urnal rotation  of  the  heavens,  it  is  necessary,  in  order  to  be  able  to 
determine  the  parallax  in  longitude  at  any  given  instant,  to  know 
the  situation  of  the  ecliptic  at  the  same  instant. 

This  is  ascertained  by  finding  the  situation  of  the  point  of  the 
ecliptic  90°  distant  from  the  points  in  which  it  cuts  the  horizon, 
and  which  are  respectively  just  rising  and  setting,  called  the  Non- 
agesimal  Degree,  or  the  Nonagesimal. 

Fig.  126.  Let  K  (Fig.  126)  be  the 

pole  of  the  ecliptic/ft,  p  the 
pole  of  the  equator  fa  ;/is 
the  vernal  equinox,  the  ori- 
gin of  longitudes  and  of 
right  ascensions  ;  hbs  is  the 
eastern  horizon,  b  the  hor- 
oscope, or  the  point  of  the 
eclipticwhich  is  just  rising; 
pz  —  90°  —  /  (the  latitude 
of  given  place)  ;  Kp  =  w  the  obliquity  of  the  ecliptic.  The  circle 
Kznv  is  at  the  same  time  perpendicular  at  n  to  the  ecliptic  fb,  and 
at  v  to  the  horizon  Jib  ;  it  is  a  circle  of  latitude  and  a  vertical  cir- 
cle, since  it  passes  through  the  pole  K  and  the  zenith  'z:  b  is  90° 
from  all  the  points  of  the  circle  Knv  ;  zn  is  the  latitude  of  the  ze- 
nith,^ its  longitude  ;  the  point  n  is  the  nonagesimal,  since  bn  = 
90°  ;  nv  is  the  altitude  of  this  point,  and  the  complement  of  zn  ; 
nv  measures  the  inclination  of  the  ecliptic  to  the  horizon  at  the 
given  instant,  or  the  angle  b,  so  that  b  —nv  =  Kz  ;  thus/ra  =N 
the  longitude  of  the  nonagesimal,  and  nv  =  h  the  altitude  of  the 
nonagesimal,  designate  the  situation  of  this  point,  and  conse- 
quently ascertain  the  position  of  the  ecliptic  and  its  pole  at  the 
moment  of  observation.* 

*  Francoeur's  Uranography,  p.  421. 


LONGITUDE  AND  ALTITUDE  OF  THE  NONAGESIMAL.  353 

The  points  m  and  d  are  those  of  the  equator  and  ecliptic  which 
are  on  the  meridian  ;  the  arc  fm,  in  time,  is  the  sidereal  time  s» 
which  is  known  ;  the  aicfi  =  90°,  since  the  plane  Kpi,  passing 
through  the  poles  K  and  p,  is  at  the  same  time  perpendicular  to 
the  ecliptic  and  to  the  equator;  the  arc  mi  =fi  —  fm  —  90°  —  s  ; 
then  the  angle  zpK  =  I80°—zpi  =  180°—  mi  =  90°+  s* 

Now,  in  the  spherical  triangle  pKz  we  know  the  sides  Kp  =  ut 
zp  =  9Q°—l  —  H,  and  the  included  angle  zpK  —  90°  +  s  ;  and 
may  therefore  find  Kz  =  h  the  altitude  of  the  nonagesimal,  and  the 
angle  pKz  =  nc  —fc—fn  =  90°—  N  =  complement  of  the  longi- 
tude N  of  the  nonagesimal.  Let  S  =  sum  of  the  angles  Kzp  and 
zKp,  then,  (For.  86,  page  346,) 


but, 

tan  iS  =-tan  (180°-£S),  and  tan  £  (90°--s)  =-tan  i  (s-90°)  ; 

substituting,  and  denoting  (180°  —  58)  by  E,  we  have 


Again,  letD  =  zKp—  Kzp,  then,  (For.  87,) 


whence,  by  transforming  as  above,  and  denoting  (180°—  £D)  by  F, 
we  have 


Now,  iS  -f  sD  =pKz  =  90°—  N  ; 

whence,  N  =  90°—  (iS  +  £D), 

or, 

N  =  360°  4-90°-  (IS  4-  £D)  =  180°-£S  +180°-iD  +90°; 

consequently,  N  —  E  -f  F  -f  90°  .  .  .  (J), 

rejecting  360°  when  the  sum  exceeds  that  number. 

Next,  for  the  altitude  of  the  nonagesimal,  we  have,  (For.  88,) 


N  and  h  being  known,  to  obtain  the  formula  for  the  parallax 
in  longitude  and  latitude,  we  have  only  to  replace  in  the  formula 


*  Francoeur*s  Uranography,  p.  421. 
45 


354  APPENDIX. 

for  the  parallax  in  right  ascension  and  decimation,  the  altitude  I  oi 
the  pole  of  the  equator  by  that  90°—  h  of  the  pole  K  of  the  eclip 
tic,  and  the  distance  im  of  the  star  s  from  the  meridian  by  the  dis- 
tance nc  to  the  vertical  through  the  nonagesimal.  Let  us  change 
then  in  formulae  (A),  (B),  (C),  (D),  (E),  (F),  and  (G),  /  into 
90°—  A,  and  q  into  fc  —  fn  —  L—  N,  L  being  the  longitude  fc  of 
the  star  s.  Besides,  d  will  become  the  distance  sK  to  the  pole  of 
the  ecliptic,  or  complement  of  the  latitude  X  =  sc.  Making  these 
substitutions,  and  denoting  the  parallax  in  longitude  by  n,  and  the 
parallax  in  latitude  by  *,  we  obtain  in  terms  of  the  apparent  longi- 
tude and  latitude, 

sin  P  sin  h  ,  .    T      ,  ,  ,      x  /T  . 

smn  =  —  .—  -j  —  (smL—  N-f-n)  .  .  .  (L), 


sin  (L—  N  +  n)/  sin  P  cos  h  \ 

-  -(cold 
)      V 
sin  P  cos  h 


,,  ,     v      sn      —         n/  sn     cos    \ 

col(d  +  *)=  -  ^—^  —  ^r-  -(cold  --  r-3  —  1   .  .  .  (M), 
sm(L—  N)      V  smd      / 

.  (N), 


sine? 


sin  *  =  sin  P  cos  h  sin 
cos  (d  +  if)  cos  (L—  N  +  in)  sin  P  sin  h 


•••«>. 


sin  P  cos  /*.,,,  v          /rix 

sui  if  =  --  sin  (d  -f  «—  y)  .  .  .  (R); 
cos  y 

and  in  terms  of  the  true  longitude  and  latitude, 

_      PsinA     .    /T       _  _  '      /Psin^V 
n  =     .          sm  (L  -  N)  +  I      .     .    I  X 
sind  \  smd  / 

sin(L-N)cos(L-N)sinl//  .  .  ,  (S), 

Pcosh   .  ./PcosAV 

*  =  --  sm  (d-y)  +  i  I  --  I  x 
cos  y  \  cos  y  / 

sm2(d—y)sml"  .  .  .  (T), 
.  tan  A  cos  (L—  N+£n) 
cos£n 

To  facilitate  the  computation,  sin  IT,  sin  tf,  and  sin  P,  in  formu- 
lae (L),  (P),  and  (R),  maybe  replaced  by  the  arcs  themselves. 

The  distance  d  of  the  star  from  the  pole  of  the  ecliptic  enters 
into  these  formulae  in  place  of  the  latitude  X. 

To  find  the  apparent  distance  d',  we  have 


MOON'S  AUGMENTED  SEMI-DIAMETER.  356 

for  the  apparent  latitude  X', 

V=X—  »•; 

for  the  apparent  longitude  I/, 

L'=L  +  II. 

The  logarithmic  formulae  given  on  page  298,  were  derived  from 
equations  (L),  (O),  and  (P),  and  the  logarithmic  formula  on  page 
299  from  equa.  (0). 

To  determine  now  the  effect  of  parallax  upon  the  apparent  di- 
ameter of  the  moon. 

Let  ACB  (Fig.  65,  p.  147)  represent  the  moon,  and  E  the  sta- 
tion of  an  observer  ;  also  let  R  =  apparent  semi-diameter  of  the 
moon,  and  D  =its  distance.  The  triangle  AES  gives 

i  -noi  AS  .       -r,  AS 

sm  AES  =  ^TFT,  or   sin  R  =  -fr- 

HjO  LJ 

At  any  other  distance  D'  we  should  have  for  the  apparent  semi- 
diameter  R', 


sin  R'      D 

whence-  sTnir  =s>- 

Thus,  if  "R/  =  moon's  apparent  semi-diameter  to  an  observer  at  the 
earth's  surface,  as  at  O  (Fig.  26,  p.  50),  R  =the  same  as  it  would 
be  seen  from  the  centre  C,  and  S  represents  the  situation  of  the 
moon, 

sin  R'  _  CS  _  sin  ZOS  _  sin  Z 
sinR  ~OS~sinZCS~sin*  ' 
But  we  have,  (see  page  350,) 

sinZ  _  (sine?  +  5)    sin  (q  -fa) 
sin  z  sin  d  sin  q 

or,  in  terms  of  the  apparent  longitude  and  latitude,  (see  page  354,) 
sinZ      *m(d  +  *)  sin  (L  —  N  +  n) 


sin  z  sin  d  sin  (L  —  N) 

Hence,      sin  R'  =  *"  R  sin  (<*  +  *)  sin  (L  -  N  +  n)         ^ 

sin  d  sin  (L  —  N) 


Aberration  in  Longitude  and  Latitude,  and  in  Right  Ascension 
and  Declination*  (See  Art.  129,  page  59.) 

Aberration  is  caused  by  the  motion  of  light  in  conjunction  with 
the  motion  of  the  earth.  Light  comes  to  us  from  the  sun  in  8m- 
17"-.8,  during  which  time  the  earth  describes  an  arc  a  =20r.44, 

*  Francoeur's  Uranography,  p.  442,  &c. 


356 


APPENDIX. 


of  its  orbilpbdin  (Fig.  127,)  supposed  circular :  p  is  the  place  of 
the  earth.  Let  us  take  any  plane  whatsoever,  which  we  will  call 
relative,  passing  through  the  star  and  the 
sun,  and  let  dd'  be  the  intersection  of  this 
plane  and  the  ecliptic,  with  which  it  makes 
an  angle  k  :  let  us  seek  the  quantity  9  by 
which  the  aberration  displaces  the  star  in 
the  direction  perpendicular  to  this  plane. 
The  question  is  to  project  on  to  a  line  per- 
pendicular to  the  relative  plane,  the  small 
constant  arc  a  which  the  earth  describes, 
this  being  the  quantity  that  the  star  is  dis- 
placed from  its  line  of  direction  in  a  direction  parallel  to  the  line 
of  the  earth's  motion,  (see  Art.  124  of  the  text:)  this  projection 
is  9,  variable  according  to  the  position  of  the  relative  plane  in  rela- 
tion to  which  it  is  estimated.  The  velocity  along  the  tangent  at 
p,  makes  with  ph  an  angle  6  =pch  =  the  arc  pd' ;  a  cos  &  is  then 
the  projection  of  this  velocity  on  the  line  ph.  The  angle  of  our 
two  planes  being  k,  this  projection  will  be  reduced  to  a  cos  6  sin 
k,  when  it  is  taken  perpendicularly  to  the  relative  plane.  Thus, 

9  =  a  sin  k  cos  6  .  .  .  (V). 

The  aberration  displaces  the  star  from  the  relative  plane  by  this 
quantity  9,  k  designating  the  inclination  of  this  plane  to  the  eclip- 
tic, and  6  the  arc  pd',  reckoned  from  JP  the  place  of  the  earth  to  d' 
the  point  of  intersection  of  these  two  planes.  Let  us  give  to  the 
relative  plane  the  positions  which  are  met  with  in  applications. 

Let  us  suppose  at  first  that  k  =  90°,  or  sin  k  ~  1  ;  the  relative 
plane  will  then  be  perpendicular  to  the  ecliptic.  Let  n  be  the  ver- 
nal equinox  ;  we  have  pd'  =  np  —  nd' ;  np  is  the  longitude  of  the 
earth,  or  180°  -f  that  O  of  the  sun  ;  nd'  is  the  longitude  /  of  the 
star ;  whence 

9  =  —  a  cos  (O  —  I). 

Now,  let  M  (Fig.  126)  be  the  true  place 
of  the  star,  M'  the  star  as  displaced  by 
aberration,  KM  is  the  circle  01  true  lati- 
tude, KM'  the  circle  of  apparent  latitude, 
and  MM'  —  9  :  this  arc  has  its  centre  C 
on  the  axis  which  passes  through  the  pole 
K  of  the  ecliptic  ;  the  longitude  of  the 
star  is  then  altered  by  the  part  OO'  of  the 
ecliptic  comprised  between  these  two 
planes  ;  and  since  OO'  is  to  the  arc  MM' 
as  the  radius  1  is  to  the  radius  CM  =  sin  KM  =  cos  latitude  X  of 
the  star,  we  have 

aberr.  in  long.  = cos  (O  —  1)  .  .  .  (W). 

cos  X 

If  the  relative  plane  is  kc,  (Fig.  129,)  perpendicular  to  the  circle 


Fig.  128. 


ABERRATION  IN  RIGHT  ASCENSION  AND  DECLINATION.         357 

of  latitude  Kcd,  the  aberration  <p 
perpendicularly  to  it,  will  be  the 
aberration  in  latitude.     Let  kd  be 
the  ecliptic,  and  o  the  earth  ;  the 
angle  k  is  measured  by  the  arc  cd 
=  X  ;  the  arc  ok  =  &  =  O  —  long, 
of  k  ;  and  as  kd  =  90°,  long,   of 
point  k  =  I  —  90°  :  substituting  in  equation  (V),  we  find 
aberr.  in  lat.  =  —  a  sin  X  sin  (O  —  1)  -  .  .  (X). 
These  aberrations  of  the  star  produce  a  small  apparent  orbit, 
which  is  confounded  with  its  projection  on  the  tangent  plane  to 
the  celestial  sphere.     Let  us  suppose  the  orbit  to  be  referred  to 
two  co-ordinate  axes  passing  through  the  true  place  of  the  star  and 
lying  in  the  tangent  plane,  of  which  one  is  parallel  to  the  plane  of 
the  ecliptic,  and  the  other  perpendicular  to  this,  or  tangent  to  the. 

circle  of  latitude  at  the  star  ;  and  let =  aberr,  in  long.,  and 

cos  X 

y  =•  aberr.  in  lat. ;  y  will  be  the  ordinate,  and  x  (the  aberr.  in  long., 
reduced  to  the  parallel  through  the  star)  the  abscissa :  we  have 

~= ^- cos  (0-0, 

cos  X  cos  X 

y  =  — -  a  sin  X  sin  (  O  —  I) ; 
or,  —  =  —  cos  (O  —  I), 

-?— =  -sin(0-J). 
flsmX 

Squaring  the  last  two  equations,  and  adding  them  together,  O  dis- 
appears, and  we  find 

y2  +  z3sm2X=:a2sinaX  .  .  .  (Y), 

whatever  may  be  the  place  of  the  earth.  Such  is  the  equation  of 
the  apparent  orbit,  which,  as  we  perceive,  is  an  ellipse  of  which 
the  semi-axes  are  a  and  a  sin  X,  and  whose  centre  is  the  true  place 
of  the  star.  When  the  star  is  at  the  pole  of  the  ecliptic,  X  =  90°, 
and  the  ellipse  becomes  a  circle  of  which  the  radius  is  a.  When 
X  =  0,  this  ellipse  is  reduced  to  an  arc  2a  of  the  ecliptic. 

To  find  the  aberration  in  right  ascension,  the  relative  plane  must 
be  perpendicular  to  the  equator.  Let  kc  be  the  equator,  (Fig.  129,) 
p  its  pole,  psd  the  relative  plane,  which  is  the  circle  of  declination 
of  the  star  s ;  kd  the  ecliptic,  o  the  earth,  k  the  vernal  equinox, 
kc  =  R,  sc  =  D.  Aberration  carries  the  star  s  out  of  the  plane 
pcd  a  distance  <p,  which  it  is  the  question  to  determine.  Equa. 
(V)  is  here 

<p  =  a  sin  d  cos  do  — a  sin  d  cos  (Jed  —  ko) 
=  a  sin  d  (cos  kd  cos  ko  -f-  sin  kd  sin  ko) 
-=  a  sin  d  cos  kd  cos  ko  +  a  sin  d  sin  kd  sin  ko 


358  APPENDIX. 

but  ko  —  long,  of  earth  =  180°  +  O  ;  we  have  also  the  angle  k  = 
the  obliquity  w  of  the  ecliptic,  and  the  right-angled  spherical  trian- 
gle kcd  gives,  by  Napier's  rules, 

cot  kd  =  cot  R  cos  w,  sin  d  sin  kd  =  sin  R. 
The  1st  equa.  multiplied  by  the  2d,  gives 

sin  d  cos  kd  =  cos  R  cos  w, 
whence        9  =  —  a  (cos  R  cos  w  cos  O  +  sir  R  sin  O). 

The  displacement  from  M  to  M'(Fig.  128)  conducts,  as  before, 
to  the  division  of  9  by  cos  D,  to  have  the  conesponding  arc  of  the 
equator  :  thus  the  aberration  in  right  ascension  is, 

u  =  —  a  sin  R  sec  D  sin  O  —  a  cos  u  cos  R  sec  D  cos  O  (Z). 

Taking  the  relative  plane  perpendicular  to  the  circle  of  declina- 
tion, we  find  for  the  aberration  in  declination, 

v  =  —  a  sin  D  cos  R  sin  O  —  a  cos  w  (tan  w  cos  D   —  sin  R  sin  D) 
cos  O  .  .  .  (a). 

These  formulae  may  easily  be  adapted  to  logarithmic  computa- 
tion : 

In  formula  (Z)  let  a  sin  R  sec  D  =  A,  and  a  cos  w  cos  R  sec 
D  =  B;  then, 

u  =  -  A  (sin  O  +-T-  cos  O)  .  .  .  (Z'). 
A. 

~  B       a  cos  u  cos  R  sec  D  _ 

Put  tan  9  =  -r-  = -— fr —  =  cos  w  cot  R  ...  (6), 

A  a  sin  R  sec  D 

and  we  shall  have 

u  —  —  A  (sin  O  -f  ^-^  cos  O) 
cos  9 

.  sin  O  cos  9  +  sin  9  cos  0 
cos  9 

5=  — sin  (O  -I-  9). 

cos  9 

Restoring  the  value  of  A,  and  taking ^  for  sec  D,  we  obtain 

asinR 

w  =  __ sm(O-f-<p)  .  .  .  (c). 

cos  D  cos  9 

The  auxiliary  arc  9  is  given  by  equation  (b) ;  it  must  be  substi 
tuted  in  equation  (c),  with  its  sign,  and  we  then  obta  n  u.  Tan 
9,  and  the  co-efficient  of  sin  (O  H-  9)  are  constant,  for  the  same  star, 
for  a  long  period  of  time,  since  these  quantities  vary  very  slowly 
with  w  and  the  precession.  Moreover,  the  co-efficient  of  sin 
(0+9)  is  the  maximum  value  of  u,  since  it  answers  to  sin 
(0  +  9)  =  1.  Thus  we  shall  be  able  to  calculate  in  advance,  for 


NUTATION  IN  RIGHT  ASCENSION  AND  DECLINATION.  359 

any  designated  star,  the  values  of  9  and  of  the  maximum  of  the  aber- 
ration in  right  ascension,  or  of  the  logarithm  of  this  maximum. 

The  results  of  these  calculations  for  50  principal  stars  are  given 
in  Table  XCI,  columns  entitled  M  and  9. 

If  in  equation  (a)  we  make  a  sin  D  cos  R  =  A',  and  a  cos  « 
(tan  u  cos  D—  sin  R  sin  D)  =  B',  we  shall  have  the  equation 

B' 

v  =-  A'  (sin  O  +  ^7  cos  O), 

in  which  A'  and  B'  are  constants.  This  equation  is  of  the  same 
form  with  equa.(Z').  We  therefore  have,  in  the  same  manner  as 
for  the  right  ascension, 

B'  _  «  cos  oj  (tan  w  cos  D  —  sin  R  sinD) 
"A?"  a  sin  D  cos  R 

a  sin  w  cos  D  —  a  cos  w  sin  R  sin  D 

a  sin  D  cos  R 
sin  w  cot  D 


cos  w  tan  R  .  .  .  (d\ 


A'  a  sin  D  cos  R 

v  =  --  -  sin  (  O  +  o  )  =  ---  -  --  x 

cos  B  cos  6 

sin(O-H)  .  .  .  (e). 

6  is  given  by  equation  (d),  and  being  substituted  in  equation  (e), 
we  shall  have  v.    6  and  the  co-efficient  of  sin    O  -M   are  constant 


for  the  same  star,  and  we  can  therefore  calculate  in  advance  the 
value  of  this  arc,  and  of  the  co-efficient,  which  is  the  maximum 
of  the  aberration  in  declination.  Columns  entitled  6  and  N,  Table 
XCI,  contain  the^  quantities  6  and  the  logarithms  of  the  maxima  of 
the  aberration  in  declination  for  50  principal  stars. 

For  convenience  in  calculation,  the  angles  9,  6,  and  the  maxima, 
M,  N,  in  Table  XCI,  have  been  rendered  positive  in  all  cases  >. 
This  has  been  accomplished  by  adding  12s-  to  9  and  6  whenever 
the  calculation  conducted  to  a  negative  value,  and  by  adding  6*-  to 
O  +  9,  or  O  +  d,  whenever  the  co-efficient  had  the  sign  —  ,  (this 
sign  being  changed  to  +  ;)  in  this  manner  the  sign  of  each  of  the 
two  factors  is  changed,  which  does  not  alter  the  sign  of  the  pro- 
duct. 

Formula  for  the  Nutation  in  Right  Ascension  and  Declination* 
(Referred  to  in  Article  148,  p.  63.) 

In  deriving  these  formulae,  we  must  begin  with  borrowing  cer- 
tain results  established  by  Physical  Astronomy.  It  has  been 
proved,  in  confirmation  of  Bradley's  conjectures,  that  the  phenom- 
ena of  nutation  are  explicable  on  the  hypothesis  of  the  pole  of  the 
earth  describing  around  its  mean  place  (that  place  which,  see  pago 

*  Wood  house's  Astronomy,  p.  357,  &c. 


360 


APPENDIX. 


61,  it  would  hold  in  the  small  circle  described  around  the  pole  of 
the  ecliptic,  were  there  no  inequality  of  precession)  an  ellipse,  in 
a  period  equal  to  the  revolution  of  the  moon's  nodes.  The  major 
axis  of  this  ellipse  is  situated  in  the  solstitial  colure  and  equal  to 
18".50  ;  it  bears  that  proportion  to  the  minor  axis  (such  are  the 
results  of  theory)  which  the  cosine  of  the  obliquity  bears  to  the 
cosine  of  twice  the  obliquity :  consequently,  the  minor  axis  will  be 
13".77. 

Let  CdA.  (Fig.  130)  represent  such  an  ellipse,  P  being  the  mean 
place  of  the  pole,  K  the  pole  of  the  ecliptic.     CDOA  is  a  circle 

Fig.  130. 


described  with  the  centre  P  and  radius  CP.  VL  is  the  ecliptic, 
Vw  the  equator,  KPL  the  solstitial  colure.  In  order  to  determine 
the  true  place  of  the  pole,  take  the  angle  APO  equal  to  the  retro- 
gradation  of  the  moon's  ascending  node  from  V  :  draw  Oi  perpen- 
dicular to  PA,  and  the  point  in  the  ellipse,  through  which  Oi 
passes,  is  the  true  place  of  the  pole.  This  construction  being  ad- 
mitted, the  nutations  in  right  ascension  and  north  polar  distance 
may,  Pp  being  very  small,  be  thus  easily  computed. 

Nutation  in  North  Polar  Distance. 
Nutation  in  N.  P.  D.  =  Ptf—  ptf.  =  Pr  =  Pp  cospPtf,  nearly, 


=  Pp  cos  (  APp  +  R  —  90°) 
=  Ppsm 
R  denoting  the  right  ascension. 


NUTATION  IN  RIGHT  ASCENSION  AND  DECLINATION.  361 

Nutation  in  Right  Ascension. 

The  right  ascension  of  the  star  tf  is,  by  the  effect  of  nutation, 
changed  from  Vw  into  Vts.     Now, 

V'fc  =  V't>  -f  VM>  +  ts,  nearly, 
whence,  Vw  -  V'ts  =  -  V'v  —  ts 

=  -  W  cos  VV'u  -  P^  sin  Pp<f  ^s 

smP<r 

in  which  expression  V'w  (=  VV  cos  VV'r)  is,  as  in  the  case  of  pre- 
cession, common  to  all  stars. 

In  order  to  reduce  farther  the  above  expression,  we  have 
pPa  =  APp  +  APrf  =  APp  +  R  -  90°, 


whence,          —  V'v  —  ts  =  —  Pp  sin  APp  cot  w 

-  Pp  sin  (APp  +  R  -  90°)  cot  N.  P.  D. 
=  —  Pp  sin  APp  cot  co  -f  Pp  cos  (APp  +  R)  cot  <*, 

5  representing  the  north  polar  distance,  and  w  the  obliquity  of  the 
ecliptic. 

But  these  forms  are  not  convenient  for  computation.     In  order 
to  render  them  convenient,  we  must,  from  the  properties  of  the  el- 
lipse, deduce  the  values  of  Pp,  and  of  the  tangent  of  APp,  and 
then  substitute  such  values  in  the  above  expressions  :  thus, 
Pp    _  sec  APp  _  cos  APO  _  cos(128—  ft)  _    cos  ft 
PO  ~  sec  APO  ~~  cos  APp         cos  APp          cos  APp' 
ft  designating  the  longitude  of  the  moon's  ascending  node  ; 

,  -n  PO  COS  ft 

whence  Pp  =  -  r-=:  —  . 

cos  APp 

A  tan  APp       pi  _  Pd  __  Pd 

tanAPO~Oi"PD"PO; 

nence,  tan  APp=        tan  APO  =         tan  (128-  -  ft) 


Now  substitute,  and  there  will  result 

The  Nutation  in  North  Polar  Distance 

=  P°  C°^  (sin  APp  cos  R  +  cos  APp  sin  R) 
cos  APp 

=  PO  (tan  APp  cos  R  cos  ft  +  cos  ft  sin  R) 
=  —  Pd  cos  R  sin  ft  +  PO  cos  ft  sin  R 
=  -  6".887  cos  R  sin  ft  +  9".250  cos  ft  sin  R  .  .  .  (/)  ; 
46 


362  APPENDIX. 

which  is  the  difference,  as  far  as  nutation  is  concerned,  between 
the  mean  and  apparent  north  polar  distance.  The  apparent  north 
polar  distance,  therefore,  must  be  had  by  adding  the  preceding 
quantity,  with  its  sign  changed,  to  the  mean. 

Nutation  in  right  ascension  =  Pd  sin  &  cot  w 
+  PO  cos  ft  cos  R  cot  5  +  Pd  sin  ft  sin  R  cot  5, 
which,  as  far  as  nutation  is  concerned,  is  the  difference  of  the  mean 
and  apparent  right  ascensions :  and,  consequently,  the  above  ex- 
pression must  be  subtracted  from  the  mean,  in  order  to  obtain  the 
apparent  right  ascension ;  or,  which  is  the  same,  must  be  added 
after  a  negative  sign  has  been  prefixed  ;  in  which  case,  we  have, 
substituting  for  PO,  Pd  their  numerical  values, 

The  Nutation  in  Right  Ascension 

=  —  6". 887  sin  ft  cot  w 
— 9".250  cos  ft  cos  R  cot  5-6".887  sin  ft  sin  R  cot  5  .  .  .  (g). 

Formulae  (/)  and  (g)  are  of  the  same  form  with  (Z)  and  (a)  for 
the  aberrations  in  right  ascension  and  declination,  and  therefore 
formulae  may  be  derived  from  them  similar  to  (c)  and  (e),  adapted 
to  logarithmic  computation.  The  quantities  corresponding  to  <p, 
M,  £,  N,  have  been  calculated  for  the  stars  in  the  catalogue  of 
Table  XC,  and  inserted  in  Table  XCI,  in  the  columns  entitled 
<p',  M',  0',  N'. 

The  Solar  Nutation  arises  from  like  causes  as  the  Lunar,  and 
admits  of  similar  formulae.  As  an  ellipse,  made  the  locus  of  the 
true  place  of  the  pole,  served  to  exhibit  the  effects  of  the  lunar 
nutation,  so  an  ellipse,  of  different,  and  much  smaller  dimensions, 
may  be  made  to  represent  the  path  which  the  true  pole  of  the 
equator  would,  by  reason  of  the  sun's  inequality  of  force  in  caus- 
ing precession,  describe  about  the  mean  place  of  the  pole.  Thus, 
in  Figure  130,  the  ellipse  AdC  will  serve  to  represent  the  locus 
of  the  pele,  when  AP  =  0".545,  Pd  =  0".500,  and  APO,  instead 
of  being  =  ft,  is  equal  to  2  0,  or  twice  the  sun's  longitude, 
taken  in  the  order  of  the  signs  ;  the  equations,  therefore,  for  the 
solar  nutation  in  north  polar  distance,  and  right  ascension,  analo- 
gous to  eqs.jf  and  g  will  be 

The  Solar  Nutation  in  North  Polar  Distance 
—  _  0".500  cos  R  sin  2  0  +  0".545  sin  R  cos  2  0  .  .  .  (h). 

The  Solar  Nutation  in  Right  Ascension 
=  —  0".500  sin  2  0  cot  w 

—  0".545  cos  2  0  cos  R  cot  <5  —  0".500  sin  2  ©  sin  R  cot  5  . .  (*). 

If  the  apparent  place  of  a  star  should  be  required  with  great 

precision,  it  would  be  necessary  to  compute  the  solar  nutations 

from  these  formulae,  and  apply  them  as  corrections  to  the  mear. 


EFFECTS  OF  OBLATENESS  OF  THE  EARTH*S  SURFACE.     363 

right  ascension  and  declination.  The  calculation  would  be  per- 
formed after  the  same  manner  as  for  the  lunar  nutation ;  but  it  is 
much  abridged  by  remarking  that  the  form  of  the  equations  is  the 
same  as  that  of  the  equations  for  the  lunar  nutation,  and  that  the 
co-efficients  are  very  nearly  the  0.075  of  those  of  the  latter  equa- 
tions. Thus  we  can  make  use  of  the  same  arcs  <p',  6',  and  log. 
maxima,  M',  N',  repeat  the  calculation  for  the  lunar  nutation, 
taking  2  O  instead  of  &,  arid  multiply  the  nutations  in  right  ascen- 
sion and  declination  thus  obtained  by  0.075.  The  results  will  be 
the  solar  nutations  required.  (See  Prob.  XX.) 

F '  trmul&for  computing  the  effects  of  the  Oblateness  of  the  Earth's 
Surface  upon  the  Apparent  Zenith  Distance  and  Azimuth  of  a 
Star*  (See  Article  162,  page  69.) 

From  the  centre  of  the  earth,  an  observer  would  see  a  star  at  I, 
Fig.  131.  (Fig.  131,)  and  would  have  V  for  his 

zenith :  fftm  the  surface  his  zenith  is 
Z,  and  he  sees  this  star  at  B  ;  IB  =p 
is  the  parallax  in  altitude ;  the  azi- 
p  muth  VZI  is  changed  into  VZB.     If 
for  a  given  time,  we  wish  to  calculate 
the  apparent  zenith  distance  BZ,  and 
the  apparent  azimuth  VZB,  we  hav^ 
first  to  resolve  the  spherical  triangle  IZP,  in  which  we  kp- 
two  sides  ZP  =  co-latitude  and  IP  =  co-declination,  ana 
eluded  hour  angle  P  ;  the  azimuth  VZI  (=  A),  and  the  a* 
(=  n)  will  thus  be  known.     But  from  the  earth's  surface,  the  v 
is  seen  at  B  :  the  azimuth  VZB  =  VZI  +IZB  =  A  +  a  ;  the  zeniu 
distance  BZ  =  n  +p,  since,  VZ  (=  i)  being  very  small,  we  have 
sensibly  IB  -f-  IZ  =  BZ.     By  reason  of  the  want  of  sphericity  of 
the  earth,  parallax  then  increases  the  true  azimuth  and  zenith 
distance  of  a  star  by  small  quantities,  a  and  p,  which  it  is  neces- 
sary to  calculate.     In  the  triangle  VIZ  we  have 

cos  IV  =  cos  i  cos  n  +  sin  i  sin  n  cos  A  =  cos  n  +  k  sin  n  ; 

making  cos  i  =  1,  sin  i  =  i,  and  i  cos  A  =  k.  Now,  k  £.  i,  and 
&  fortiori  cos  k  =  1,  sin  A;  =  k  ;  whence 

cos  IV  =  cos  n  cos  k  •}-  sin  n  sin  k  =  cos  (n  —  A;), 
and  IV  =  n  —  k  =n  —  i  cos  A. 

Thus  we  correct  the  calculated  arc  n  by  the  quantity  —  i  cos 
A,  to  have 

IV  =  z  —  n  —  i  cos  A  ...  (j). 
If  this  value  of  z  be  introduced  into  equation  (10),  page  52,  we 

*  Francceur's  Uranography,  p.  426,  &c. 


564 


APPENDIX. 


shall  have  p,  and  thence  the  apparent  zenith  distance  Z  =  n  +  p 
=  BZ. 

Afterwards,  to  obtain  IZB  =  a,  or  the  parallax  in  azimuth,  the 
triangles  ZBV,  ZBI  give 

sin  ZBV  __  sin  (A  +  a)        sin  ZBV  ^sina 

sin  i          sin  (z  +p) '          sin  TI          sinp  ' 
whence,  by  equating  the  values  of  sin  ZBV, 

sin  n  sin  a sin  i  sin  (A  +  a) 

sin  p  sin  (z  +  p) 

substituting  for  sin  p  its  value  sin  H  sin  (z  +jp)  =  sin  H  sin  Z, 
(equa.  8,  page  51,)  and  reducing,  we  have 

sin  a       _  sin  (A  +  a) 
sin  H  sin  i  sin  n      ' 

and  as  i  is  very  small,  sin  i  sin  (A  +  a)  does  not  differ  sensibly 
from  i  sin  A,  and  we  thus  have  in  seconds,  (For.  47,  page  343,) 

Hisjn  A  sin  1" 
sin  n 


Solution  of  Kepler's  Problem,  by  which  a  Body's  Place  is  found 
in  an  Elliptical  Orbit*    (See  Art.  268,  p.  106.) 

Let  APB  (Fig.  132)  be  an  ellipse,  E  the  focus  occupied  by  the 
sun,  round  which  P  the  earth  or  any  other  planet  is  supposed  to 
revolve.  Let  the  time  and  planet's  motion  be  dated  from  the  ap- 

Fig.  132. 

M 


side  or  aphelion  A.  The  condition  given  is  the  time  elapsed  from 
the  planet's  quitting  A;  the  result  sought  is  the  place  P ;  to  be 
determined  either  by  finding  the  value  of  the  angle  AEP,  or  by 

*  Woodhouse's  Astronomy,  p.  457,  &c. 


365 

cutting  off,  from  the  whole  ellipse,  an  area  AEP  bearing  the  same 
proportion  to  the  area  of  the  ellipse  which  the  given  time  bears  to 
the  periodic  time. 

There  are  some  technical  terms  used  in  this  problem  which  we 
will  now  explain. 

Let  a  circle  AMB  be  described  on  AB  as  its  diameter,  and  sup- 
pose a  point  to  describe  this  circle  uniformly,  and  the  whole  of  it 
in  the  same  time  as  the  planet  describes  the  ellipse  ;  let  also  t  de- 
note the  time  elapsed  during  P's  motion  from  A  to  P  ;  then  if  AM  = 

—  r-j  x  2  AMB,  M  will  be  the  place  of  the  point  that  moves 

uniformly,  while  P  is  that  of  the  planet;  the  angle  ACM  is 
called  the  Mean  Anomaly,  and  the  angle  AEP  is  called  the  True 
Anomaly. 

Hence,  since  the  time  (t)  being  given,  the  angle  ACM  can  al- 
ways be  immediately  found,  (see  Art.  267,  p.  106,)  we  may  vary 
the  enunciation  of  Kepler's  problem,  and  state  its  object  to  be  the 
finding  of  the  true  anomaly  in  terms  of  the  mean. 

Besides  the  mean  and  true  anomalies,  there  is  a  third  called  the 
Eccentric  Anomaly,  which  is  expounded  by  the  angle  DC  A,  and 
which  is  always  to  be  found  (geometrically)  by  producing  the  ordi- 
nate  NP  of  the  ellipse  to  the  circumference  of  the  circle.  This 
eccentric  anomaly  has  been  devised  by  mathematicians  for  the 
purposes  of  expediting  calculation.  It  holds  a  mean  place  between 
the  two  other  anomalies,  and  mathematically  connects  them.  There 
is  one  equation  by  which  the  mean  anomaly  is  expressed  in  terms 
of  the  eccentric  ;  and  another  equation  by  which  the  true  anomaly 
is  expressed  in  terms  of  the  eccentric. 

We  will  now  deduce  the  two  equations  by  which  the  eccentric 
is  expressed,  respectively,  in  terms  of  the  true  and  mean  anomalies. 
Let  t   =•  time  of  describing,  AP, 

P  =  periodic  time  in  the  ellipse, 

a  =  CA, 

ae  =  EC, 

v  =  L  PEA, 

u  =  L  DCA  ;  (whence,  ET,  perpendicular  to  DT,  =  EC 
x  sin  M,) 

P   =PE, 

if  =  3.14159,  &c.  ; 

then,  by  Kepler's  law  of  the  equable  description  of  areas, 


px_=px 

area  of  elhp.  area  circle       *a?  v 

P  /ET.DC      AD.DC 


P  Pi 

=  —  (e  sin  u  +  u)  :  hence,  if  we  put  —  =  -, 

*flt  tiflf         71 


366  APPENDIX. 

we  have 

nt  =  e  sin  u  +  u  .  .  .  (/), 

an  equation  connecting  the  mean  anomaly  nt,  and  the  eccentric  u. 
In  order  to  find  the  other  equation,  that  subsists  between  the 
true  and  eccentric  anomaly,  we  must  investigate,  and  equate,  two 
values  of  the  radius-vector  p,  or  EP. 

First  value  of  p,  in  terms  of  v  the  true  anomaly, 


== 


1  —  e  cost; 
Second,  in  terms  of  u  the  eccentric  anomaly, 

p  =  a   (1  +  e  cos  u)  .  .  .  (2). 
For,  P2  =  EN2  +  PN2 

=  EN2  +  DN2  x  (1  -  e2) 

—   (ae  +  a  cos  u)2  +  a2  sin2  u  (1  —  e2) 

=  a2  |e*+  2e  cos  u  +  cos2  u}  +  a2  (1  —  e2)  sin2  u 

=  a2  \  I  +  2e  cos  u  +  e2  cos2  M^  . 

Hence,  extracting  the  square  rooj, 

p  =  «  (  1  +  e  cos  #). 
Equating  the  expressions  (1),  (2),  we  have 

(1  —  e2)  =  (1  —  e  cos  v)  (1  -he  cos  M),  whence, 

e  +  cos  u 

cosv  =  7—:  --  ,  an  expression  for  v  in  terms  of  u  ; 
1  +  e  cos  u 

but,  in  order  to  obtain  a  formula  fitted  to  logarithmic  computation, 
we  must  find  an  expression  for  tan  -  :  now,  (see  For.  12,  p.  341,) 

v  _       //I  —  cos  v\  _        /  /(I  —  e)  (I  —  cos  u)\ 
111  2  "  V  VI  +  oo*  */  =  "  V  V(l  +e)l  +COSM/ 


These  two  expressions  (Z)  and  (m),  that  is, 


analytically  resolve  the  problem,  and,  from  such  expressions,  by 
certain  formulae  belonging  to  the  higher  branches  of  analysis,  may 
v  be  expressed  in  the  terms  of  a  series  involving  nt. 

Instead,  however,  of  this  exact  but  operose  and  abstruse  method 
of  solution,  we  shall  now  give  an  approximate  method  of  express- 
ing the  true  anomaly  in  terms  of  the  mean. 

MO  is  drawn  parallel  to  DC.     (1.)  Find  the  half  difference  of 


367 

the  angles  at  the  base  EM  of  the  triangle  ECM,  from  this  ex- 
pression, 

tan  i  (CEM  -  CME)  =  tan  J  (CEM  +  CME)  x  j-=-|, 

in  which,  CEM  +  CME  =  ACM,  the  mean  anomaly. 

(2.)  Find  CEM  by  adding  J  (CEM  +  CME)  and  J  (CEM 
—  CME)  and  use  this  angle  as  an  approximate  value  to  the  ec- 
centric anomaly  DC  A,  from  which,  however,  it  really  differs  by 
L  EMO. 

(3.)  Use  this  approximate  value  of  L  DC  A  =  L  ECT  in 
computing  ET  which  equals  the  arc  DM  ;  for,  since  (see  p.  365), 

t  = : — ;-  x  DE  A,  and  (the  body  being  supposed  to  revolve  in 

area  circle  J 

P 

the  circle  ADM)  =  — ;     —j-  x  ACM5  area  AED  =  area  ACM, 
area  circle 

or,  the  area  DEC  +  area  ACD  =  area  DCM  +  area  ACD  ;  con- 
sequently the  area  DEC  =the  area  DCM,  and,  expressing  their 
values, 

ET  x  DC     DM  x  DC       ,  ,       __,      ^ 

= ,  and  thus,  ET  =  DM. 

2  «£ 

Having  then  computed  ET  =  DM,  find  the  sine  of  the  resulting 
arc  DM,  which  sine  =  OT ;  the  difference  of  the  arc  and  sine 
(ET  -  OT)  gives  EO. 

(4.)  Use  EO  in  computing  the  angle  EMO,  the  real  difference 
between  the  eccentric  anomaly  DC  A  and  the  /.  MEC  ;  add  the 
computed  L  EMO  to  L  MEC,  in  order  to  obtain  L  DCA.  The 
result,  however,  is  not  the  exact  value  of  L  DCA,  since  L  EMO 
has  been  computed  only  approximately ;  that  is,  by  a  process  which 
commenced  by  assuming  L  MEC  for  the  value  of  the  L  DCA. 

For  the  purpose  of  finding  the  eccentric  anomaly,  this  is  the 
entire  description  of  the  process ;  which,  if  greater  accuracy  be 
required,  must  be  repeated  ;  that  is,  from  the  last  found  value  of 
L  DCA  =  L  ECT,  ET,  EO,  and  L  EMO  must  be  again  com 
puted. 


369 


NOTE   1. 

The  number  of  planets  known  at  the  present  date  (January  1st,  1852),  is 
twenty- two.  During  the  last  seven  years  twelve  new  planets  have  been  dis- 
covered. The  following  table  contains  the  names  of  these  planets,  together 
with  the  date  and  place  of  discovery,  and  the  name  of  the  discoverer. 


Names. 

When  discovered. 

By  ^hom. 

Where. 

Astraea 

Dec.      8,  1845 

Hencke 

Driessen. 

Neptune 
Hebe 

Sept.  23,  1846 
July     1,  1847 

Galle 
Hencke 

Berlin. 
Driessen. 

Iris 

Aug.  13,  1847 

Hind 

London. 

Flora 

Oct.    18,  1847 

Hind 

London. 

Metis 

April  25,  1848 

Graham 

Markree. 

Hygeia 

April  12,  1849 

Gasparis 

Naples, 

Parthenope 
Clio 

May    13,  1850 
Sept.  13,  1850 

Gasparis 
Hind 

Naples. 
London. 

Egeria 

Nov.     2,  1850 

Gasparis 

Naples. 

Irene 

May   20,  1851 

Hind 

London. 

Eunomia 

July   29,  1851 

Gasparis 

Naples. 

Although  Neptune  was  first  seen  by  Galle,  at  Berlin,  the  honor  of  the  discov- 
ery of  this  planet  is  generally  awarded  to  Leverrier,  a  French  astronomer. 
Leverrier  ascertained,  from  a  careful  examination  of  the  motions  of  Uranus,  that 
that  planet  must  be  subject  to  the  disturbing  action  of  an  unknown  planet  more 
remote  from  the  sun.  He  investigated  the  probable  orbit  and  mass  of  this 
unknown  planet,  that  is,  the  orbit  and  mass  that  would  serve  to  account  for  the 
previously  unexplained  irregularities  observed  in  the  motions  of  Uranus,  and 
assigned  its  probable  place  in  the  heavens.  At  his  request  Galle,  of  the  Berlin 
Observatory,  undertook  the  search  for  it ;  and  on  directing  his  telescope  to  the 
part  of  the  heavens  designated  by  Leverrier,  detected  the  supposed  planet 
within  1°  of  the  place  which  had  been  assigned  by  that  astronomer. 

The  same  investigation  was  undertaken  about  the  same  time,  and  with  very 
nearly  the  same  results,  by  a  young  English  mathematician  by  the  name  of 
Adams,  who  is  therefore  entitled  to  a  share  of  the  honor  of  this  wonderful 
discovery. 

The  planets  Ceres,  Pallas,  Juno,  and  Vesta,  on  account  of  their  diminutive 
size  and  certain  other  peculiarities,  have  received  the  appellation  of  Asteroids. 
All  the  newly-discovered  planets,  with  the  exception  of  Neptune,  are  also  classed 
among  the  asteroids.  The  number  of  asteroids  at  present  known  is,  accordingly, 
fifteen.  "  Besides  these  fifteen,  others  yet  undiscovered  may  exist ;  and  it  ia 
extremely  probable  that  such  is  the  case, — the  multitude  of  telescopic  stars 
being  so  great  that  only  a  small  fraction  of  their  number  has  been  sufficiently 
noticed  to  ascertain  whether  they  retain  the  same  place  or  not,"  and  from  one 
to  three  new  asteroids  having  been  discovered  every  year  since  1846. 

47 


370 


NOTES  n.  m. 


II. 


At  the  present  date  (Jan.,  1852),  the  largest  and  best  telescope  in  the  United 
States  is  the  great  refractor  at  the  Cambridge  Observatory,  manufactured  by 
Merz  and  Mahler,  of  Munich,  Bavaria.  The  aperture  of  the  object-glass  is  15 
inches,  and  its  focal  length  is  22£  feet.  It  has  18  different  powers,  varying  from 
180  to  2,000.  Its  dimensions  are  a  trifle  greater  than  those  of  the  Pulkova 
refractor,  and  it  is  generally  conceded  to  be  superior  to  it  in  its  performance. 
It  is,  accordingly,  the  best  refracting  telescope  in  the  world.  It  was  erected  in 
June,  1847,  and  in  the  hands  of  Messrs.  W.  C.  and  G.  P.  Bond  has  already 
enriched  astronomy  with  many  valuable  observations  and  discoveries. 

The  accuracy  of  transit  observations  has  recently  been  greatly  increased  by 
the  introduction  of  the  Electro-Chronograph  ;  by  which,  with  the  adaptation  of  a 
proper  electro-magnetic  recording  apparatus,  the  seconds  measured  off  by  the  pen- 
dulum of  a  clock  are  designated  by  a  series  of  equally  distant  dots  or  breaks  in  a 
continuous  line,  upon  a  fillet  or  roll  of  paper  to  which  an  equable  motion  is  given 
by  machinery.  The  observer  holds  in  his  hand  a  break  -circuit  key,  by  means  of 
which  he  interrupts  the  circuit  at  the  instant  that  the  star  is  bisected  by  one  of 
the  wires  in  the  field  of  the  telescope,  and  thus  makes  a  break  in  one"  of  the 
short  lines  on  the  fillet,  that  designate  the  duration  of  the  successive  seconds. 
In  this  way  it  is  believed  that  the  instant  of  the  transit  across  a  single  wire  can 
be  noted  to  within  a  much  smaller  fraction  of  a  second  than  by  the  common 
method.  Besides,  the  number  of  bisections  in  a  single  culmination  of  a  star,  by 
increasing  the  number  of  wires,  may  be  multiplied  some  seven-fold. 

This  method  of  observation  has  been  introduced  at  the  Cambridge  Observa- 
tory, and  also  at  the  National  Observatory. 


NOTE   III. 


ELEMENTS  OF  THE   ORBITS   OF  THE   ASTEROIDS, 
Arranged  in  the  Order  of  their  Mean  Distance  from  the  Sun. 


Name. 

Distance. 

Period  in 

days. 

Eccentricity. 

Inclination. 

Longitude  of 
Ascending  Node. 

o     /         // 

Q           1           11 

1 

Flora 

2.201687 

1193.249 

.156557 

5  43     4.8 

110  18  12.0 

2 

Clio 

2.334876 

1303.127 

.217922 

8  23     1.9 

235  19  49.8 

3 

Vesta 

2.361081 

1325.147 

.089569 

7     8  29.7 

103  23  31.6 

4 

Iris 

2.380624 

1341.636 

.229942 

5  28  15.9 

259  48  10.2 

5 

Metis 

2.385607 

1345.850 

.120253 

5  34  27.8 

68  32  17.4 

6 

Eunomia 

2.399440 

1357.573 

.136504 

13     0  18.5 

292  51     1.8 

7 

Hebe 

2.425786 

1379.994 

.200180 

14  47  56.0 

138  29  42.6 

8 

Parthenope 

2.450833 

1401.000 

.099466 

4  36  56.7 

124  57  55.8 

9 

Irene 

2.552303 

1518.943 

.170022 

8  37  35.7 

87  47  46.2 

10 

Egeria 

2.560070 

1492.230 

.096180 

15  57  59.8 

43  85  24.4 

11 

Astraea 

2.577047 

1511.095 

.188058 

5  19  22.7 

141  25  14.6 

12 

Juno 

2.670837 

1594.296 

.254884 

13     3  22.1 

170  54  45.6 

18 

Ceres 

2.768051 

1682.125 

.076652 

10  37     4.4 

80  48  66.6 

14 

Pallas 

2.772858 

1686.510 

.239815 

34  37  33.0 

172  43  59.7 

15 

Hygeia 

8.150060 

2042.101 

.010103 

3  47  15.5 

287  37     8.6 

NOTE   IV. 


371 


Name. 

Longitude  of 
Perihelion. 

Mean  Anomaly 
at  Epoch. 

Epoch  in  Mean  Time. 

O             /             // 

o       /         // 

d.      h. 

1 

Flora 

33     0  40.8 

35  48     7.0 

Berlin  M.  T.  1848,  Jan.      1     0 

2 

Clio              |302  55     1.5 

65  47  23. 

1851,  Jan.       0     0 

3 

Vesta 

250  46  32.2 

225  44  18.8 

"               1850,  Jan.      9     0 

4 

Iris 

41  41   13.5 

330  41  54. 

"               1848,  Jan.       1     0 

5 

Metis 

70  33  42.8 

146  30  18.5 

1848,  May     5  12 

6 

Eunomia 

112  18  15.6 

172  10  21.6 

"               1851,  Aug.     5     0 

7 

Hebe 

14  50  50.3 

275     8  51.3 

«               1847,  Jan.       1     0 

8 
9 

Parthenope 
Irene 

316  49  61.8 
191     8  27.5 

288  40  43.2 
41  57     9.5 

"               1850,  May    25     0 
Greenwich     ]  851,  June   10     0 

10 

Egeria 

116  26  49.4 

288  37  17. 

1850,  Nov.      2    0 

11 

Astraea 

135  20  47. 

318  45     3.3 

Berlin             1846,  Jan.       1     0 

12 

Juno 

54  24  12.8 

124  31  10.8 

1850,  April    8     0 

13 

Ceres 

147  46  12.4 

219     6  29.5 

"               1850,  Sept.  25     0 

14 

Pallas 

121  21  48.5 

217  31  10.6 

1850,  Aug.   23     0 

15 
—  »  — 

Hygeia 

227  49  54.2 

330  52     8.5 

1849,  April  15     0 

NOTE   IV. 

The  number  of  planets  which  are  now  known  to  have  the  situations  men- 
tioned in  the  text  is  no  less  than  fifteen.  It  is  a  remarkable  fact,  with  respect 
to  these  asteroids,  as  they  are  called,  that  their  orbits,  if  we  except  those  ot 
Iris  and  Hygeia,  have  approximately  two  common  points  of  reunion  in  opposite 
regions  of  the  heavens.  This  singular  fact  is  in  accordance  with  a  theory  pro- 
pounded by  Dr.  Olbers  nearly  fifty  years  ago  (1802),  after  the  discovery  of 
Ceres  and  Pallas,  that  "  these  small  bodies  were  merely  the  fragments  of  a  larger 
planet,  which  had  exploded  from  some  internal  convulsion,  and  that  several  more 
might  yet  be  discovered."  For,  since  the  supposed  fragments  must  have  origi- 
nally diverged  from  the  same  point,  their  paths  must,  agreeably  to  the  laws  ot 
planetary  motions,  have  two  common  points  of  reunion ;  viz.,  the  place  occupied 
by  the  primitive  planet  at  the  time  when  the  convulsion  occurred,  and  the  point 
in  the  heavens  diametrically  opposite  to  this.  It  is  true  that,  as  a  matter  of 
fact,  the  intersection  is  only  approximate,  the  deviations  from  a  common  point 
being  in  some  instances  as  much  as  4°,  and  in  the  case  of  the  planetoids  Iris  and 
Hygeia  no  less  than  9°,  but  this  discrepancy  is  ascribed,  by  the  advocates  of 
Giber's  theory,  to  the  disturbing  actions  of  the  planets,  and  the  consequent  sec- 
ular displacement  of  the  orbits  of  the  asteroids,  and  it  is  accordingly  conjectured 
that  if  the  secular  motion  of  the  node  of  each  orbit  were  known,  we  might,  by 
calculating  back,  find  that  at  some  period  in  the  past  the  orbits  all  had  truly  a 
common  point  of  intersection,  and  thus  determine  the  date  of  the  supposed  ex- 
plosion of  the  single  primeval  planet.  On  this  point  Professor  Loomis  remarks 
that  "  we  may  safely  assume  that  the  nodes  of  all  the  asteroids  have  not  coin- 
cided within  a  period  of  many  thousand  years ;  and  therefore  that,  if  these  bodies 
are  the  fragments  of  a  larger  planet  which  has  exploded,  this  explosion  must 
have  taken  place  at  a  very  remote  epoch. 

"  It  should  also  be  observed,  that  not  only  must  the  nodes  of  all  the  asteroids 
coincide,  but  the  distance  of  the  planets  from  the  sun  must  be  the  same  at  that 
instant.  Now  the  distance  of  these  planets  from  the  sun  when  at  their  nodes, 
varies  by  nearly  the  radius  of  the  earth's  orbit ;  so  that  to  bring  them  all  to- 
gether, we  must  suppose  a  corresponding  change  in  the  place  of  their  perihe- 
lia. This  also  would  require  the  lapse  of  many  centuries ;  and  when  we  con- 
sider the  necessity  of  a  coincidence  at  the  same  instant,  both  in  distance  and  di- 
rection, we  can  easily  suppose  that  such  a  result  could  not  have  taken  place 
within  a  million  of  years." 


372  NOTE  V. 

4 

NOTE   V. 

Gambart's  or  Biela's  comet,  at  its  return  in  1846,  exhibited  a  phenomenon  al- 
together unprecedented  in  the  annals  of  astronomy.  On  the  13th  of  January,  at 
the  National  Observatory  in  Washington,  and  on  the  15th  and  subsequently,  at 
all  the  principal  observatories  in  this  country  and  Europe  it  was  distinctly  seen 
to  have  become  double  ;  a  very  small  and  faint  cometic  body,  having  a  nucleus  ot 
its  own,  being  observed  appended  to  it  at  a  distance  of  about  2'  from  its  centre. 
The  two  comets  moved  on  side  by  side,  for  a  period  of  two  months,  and  through 
an  arc  of  more  than  70°,  when  the  companion,  after  undergoing  remarkable 
changes  of  magnitude  and  luminosity,  disappeared.  During  the  whole  of  this 
interval  the  apparent  distance  between  the  two  bodies  gradually  increased,  but 
the  apparent  direction  of  the  line  of  junction  remained  nearly  the  same.  On 
the  30th  of  January,  the  distance  of  separation  had  increased  to  3',  on  the  13th 
of  February  to  5',  and  so  until  on  the  5th  of  March  it  was  over  9'.  "  Both  bodies 
had  nuclei,  both  had  short  tails,  parallel  in  direction,  and  nearly  perpendicular 
to  the  line  of  junction;  but  whereas,  at  its  first  observation  on  January  13th, 
the  new  comet  was  extremely  small  and  faint,  in  comparison  with  the  old,  the 
difference,  both  in  point  of  light  and  apparent  magnitude,  diminished.  On  the 
10th  of  February,  they  were  nearly  equal,  although  the  day  before  the  moon- 
light had  effaced  the  new  one,  leaving  the  other  bright  enough  to  be  well  ob- 
served. On  the  14th  and  16th,  however,  the  new  comet  had  gained  a  decided 
superiority  of  light  over  the  old,  presenting  at  the  same  time  a  sharp  and  star- 
like  nucleus,  compared  by  Lieut.  Maury  to  a  diamond  spark.  But  this  state  of 
things  was  not  to  continue.  Already,  on  the  18th,  the  old  comet  had  regained 
its  superiority,  being  nearly  twice  as  bright  as  its  companion,  and  offering  an'un- 
usually  bright  and  starlike  nucleus.  From  this  period  the  new  companion  began 
to  fade  away,"  but  continued  visible  until  after  the  middle  of  March.  As  seen 
by  the  author  on  the  17th  of  March  in  a  reflecting  telescope  of  14  ft.  focus, 
with  a  low  power,  the  cometic  mass  had  two  points  of  maximum  brightness,  but 
the  twin  comets  were  not  distinctly  separate.  On  March  21  it  appeared  in  the 
same  telescope  as  one  nebulous  mass,  with  a  single  point  of  concentration.  On 
the  22d  of  April  this  had  disappeared. 

"  While  this  singular  interchange  of  light  was  going  forward,  indications  of 
some  sort  of  communication  between  the  comets  were  exhibited.  The  new  or 
companion  comet,  besides  its  tail,  extending  in  a  direction  parallel  to  that  of  the 
other,  threw  out  a  faint  arc  of  light  which  extended  as  a  kind  of  bridge  from 
the  one  to  the  other ;  and  after  the  restoration  of  the  original  comet  to  its  former 
pre-eminence,  it,  on  its  part,  threw  forth  additional  rays,  so  as  to  present  (on  the 
22d  and  23d  of  February,  as  seen  by  Lieut.  Maury,  of  the  National  Observatory) 
the  appearance  of  a  comet  with  three  faint  tails  forming  angles  of  about  120° 
with  each  other,  one  of  which  extended  towards  its  companion." 

What  was  the  relation  of  these  two  bodies  ?  Was  the  original  comet  actually 
divided  into  two,  as  appearances  seemed  to  indicate?  Professor  Plantamour, 
director  of  the  observatory  of  Geneva,  has  furnished  a  partial  answer  to  these 
questions.  He  has  found  that  all  the  observations  are  very  well  represented 
by  supposing  that  each  nucleus  described  an  independent  ellipse  around  the 
sun.  He  has  computed  the  orbits  of  the  two  bodies  upon  this  supposition,  from 
the  extensive  and  careful  series  of  observations  made  upon  them,  and  taking 
into  account  the  disturbing  influence  of  Jupiter,  Mars,  the  Earth,  and  Venus ; 
and  concludes  that  "  the  disturbing  action  of  one  nucleus  upon  the  other  must 
have  been  extremely  small,  and  that  it  is  doubtful  whether  the  observations 
were  sufficiently  precise  to  render  this  influence  in  any  degree  sensible.  He 
has  also  shown  that  the  increase  of  distance  between  the  two  nuclei,  at  least 
during  the  interval  from  February  IQth  to  March  22,  was  simply  apparent,  be- 
ing due  to  the  variation  of  distance  from  the  earth  and  to  the  angle  under  which 
their  line  of  junction  presented  itself  to  the  visual  ray  ;  the  real  distance  during 
all  that  interval  (neglecting  small  fractions)  having  been  on  an  average  about 
thirty -nine  times  the  semi-diameter  of  the  earth,  or  less  than  two-thirds  the  dis- 
tance of  the  moon  from  the  earth's  centre." 

If  it  be  true  that  the  two  bodies  are  in  no  sensible  degree  disturbed  by  their 
mutual  actions,  as  M.  Plantamour  infers  from  bis  investigations,  and  as  we  should 


NOTES  VI-IX. 

naturally  suppose  from  the  probable  minuteness  of  the  two  oometary  masses,  it 
has  been  calculsted  by  Sir  John  Herschel,  from  Plantamour's  elements,  that 
there  will  be  an  interval  of  16d-  4  between  their  next  perihelion  passages;  "  and 
it  will  be  therefore  necessary,  at  their  next  reappearance,  to  look  out  for  each 
comet  as  a  separate  and  independent  body."  "  Nevertheless,"  as  remarked  by 
Herschel,  "  as  it  is  still  perfectly  possible  that  some  link  of  connection  may  sub- 
sist between  them,  it  will  not  be  advisable  to  rely  on  this  calculation  to  the 
neglect  of  a  meet  vigilant  search  throughout  the  whole  neighborhood  of  the 
more  conspicuous  one,  lest  the  opportunity  should  be  lost  of  pursuing  to  its  con- 
clusion the  history  of  this  strange  occurrence." 

The  investigations  of  M.  Plantamour  have  served  to  establish  that  the  actual 
separation  of  the  two  bodies  did  not  occur  at  the  time  of  the  apparent  separa- 
tion in  1846.  At  what  point  of  time  anterior  to  that  epoch  it  took  place,  it 
would  seem  to  be  impossible  to  determine.  In  fact,  it  is  quite  possible  that  the 
two  bodies  have  been  revolving  independently  of  each  other  for  an  indefinite 
time,  and  that  the  supposed  division  of  one  comet  into  two  was  really  the  chance 
approach  of  two  independent  cometary  bodies.  Plantamour  remarks  that  "  the 
extraordinary  changes  which  the  companion  exhibited  within  the  period  of  » 
few  days,  and  which  have  often  been  noticed  in  other  comets,  seem  to  indicate 
that  the  brightness  of  these  objects  does  not  depend  merely  upon  their  distance 
from  the  earth  and  sun,  but  upon  other  unknown  causes.  These  causes  might 
have  developed  sufficient  brightness  in  the  companion  at  its  late  return  to  the 
sun  to  render  it  visible  to  us ;  while  at  its  former  returns,  on  account  of  its  un- 
favorable position,  the  companion  was  too  faint  to  be  noticed," 


NOTE   VI. 

The  list  given  in  the  text  has  recently  been  increased  by  the  addition  of  several 
other  comets,  viz.,  De  Vice's  comet,  period  6^-  years,  perihelion  passage  Sept.  2d, 
1844  ;  Brorsen's  comet,  period,  according  to  Hind,  5|  years,  perihelion  passage 
Feb.  25th,  1847  ;  Peters'  comet,  period  nearly  16  years,  perihelioa  passage  June 


NOTE   VII. 

The  reader  will  find  a  <somplete  catalogue  of  all  comets  whose  orbits  have  been 
determined,  up  to  1846,  m  the  American  Almanac  for  184*7. 


NOTE   VIII 

The  new  planet,  Neptune,  proves  to  be  the  third  planet  in  the  order  of  mag- 
nitude, being  a  little  larger  than  Uranus.  The  newly-discovered  asteroids 
are  probably  of  a  more  diminutive  size  than  the  other  four. 


NOTE   IX. 

A  remarkable  analogy  in  the  periods  of  rotation  of  the  primary  planets  was 
discovered  a  few  years  since  (1848)  by  Daniel  Kirkwood,  of  Pottsville,  Pennsyl- 
vania. This  analogy  is  now  generally  known  by  the  name  of  JKirkwood's  LOAD, 
and  is  as  follows : 

"  Let  P  be  the  point  of  equal  attraction  between  any  planet  and  the  one  next 


374  NOTE  x. 

interior,  the  two  being  in  conjunction :  P'  that  between  the  same  and  the  on* 
next  exterior. 

Let  also  D  =  the  sum  of  the  distances  of  the  points  P,  P'  from  the  orbit  of  the 
planet ;  which  I  shall  call  the  diameter  of  the  sphere  of  the  planet's  attraction ; 

D7=the  diameter  of  any  other  planet's  sphere  of  attraction  found  in  like 
manner : 

n  =  the  number  of  sidereal  rotations  peformed  by  the  former  during  one  side- 
real revolution  round  the  sun ; 

n'=  the  number  performed  by  the  latter ;  then  it  will  be  found  that 

n2  :  w'2  : :  D3  :  D'3;  or  n  =  n'  (p 

That  is,  the  square  of  the  number  of  rotations  made  by  a  planet  during  one  revo- 
lution round  the  sun,  is  proportional  to  the  cube  of  the  diameter  of  its  sphere  of 

attraction  ;  or  —  is  a  constant  quantity  for  all  the  planets  of  the  solar  system. 

The  analogy  thus  announced  has  been  subjected  to  a  rigid  mathematical  ex- 
amination by  Mr.  Sears  C.  Walker,  with  the  following  result :  "  We  may  there- 
fore conclude,"  says  he,  "that  whether  Kirkwood's  Analogy  is  or  is  not  the  ex- 
pression of  a  physical  law,  it  is  at  least  that  of  a  physical  fact  in  the  mechanism 
of  the  universe."  (See  the  American  Journal  of  Science,  New  Series,  vol.  x. 
pp.  19-26.) 

There  are  but  three  planets,  viz.,  Venus,  the  Earth,  and  Saturn,  for  which  all 
the  elements  embraced  in  this  law  are  known.  The  diameters  of  the  spheres  of 
attraction  of  Mercury  and  Neptune  are,  from  the  nature  of  the  case,  incapable 
of  determination.  The  mass  of  the  one  planet  into  which  the  asteroids  are  sup- 
posed once  to  have  been  united  is  not  known  with  certainty,  as  there  may  be 
asteroids  yet  undiscovered,  and  its  period  of  rotation  is  hypothetical  only.  The 
diameters  of  the  spheres  of  attraction  of  Mars  and  Jupiter  can  only  be  approx- 
imately determined;  and  the  period  of  rotation  of  Uranus  is  unknown.  Pro- 
fessor Loomis,  in  a  recent  article,  argues  with  a  good  deal  of  plausibility,  that 
"  Uranus  and  the  asteroids  cannot  be  reconciled  with  Kirkwood's  Law  by  any 
admissible  assumption  with  regard  to  the  value  of  their  elements."  (See  Silll- 
man's  Journal,  vol.  xl  p.  217.) 

The  objections  urged  by  Professor  Loomis  have  been  answered  by  Professor 
Kirkwood.  (See  the  Journal  of  Science,  Second  Series,  vol.  XL  p.  394.)  The 
considerations  adduced  by  him  have  served  materially  to  weaken  the  force  of 
these  objections. 

The  interest  naturally  awakened  by  the  announcement  of  so  important  a  dis- 
covery was  heightened  by  the  fact,  that  it  was  at  once  perceived  that  it  furnished 
a  new  and  powerful  argument  in  support  of  the  nebular  hypothesis  (or  cosmog- 
ony) devised  by  Laplace.  (See  a  paper  on  this  subject  by  Dr.  B.  A.  Gould,  Jr., 
in  the  Journal  of  Science,  New  Series,  vol.  x.  p.  26,  <fcc.) 


NOTE   X. 

A  new  ring  of  Saturn,  interior  to  the  other  two,  was  discovered  by  Mr.  G.  P. 
Bond,  assistant  at  the  observatory  of  Harvard  University,  on  the  llth  of 
November,  1850.  It  was  subsequently  observed  by  the  Messrs.  Bond  on  re- 
peated occasions,  from  that  date  to  the  7th  of  January,  1851.  It  shone  with  a 
pale  dusky  light.  Its  inner  edge  was  sharply  defined,  but  the  side  next  the  old 
ring  was  not  so  definite  ;  so  that  it  was  impossible  to  make  out  with  certainty 
whether  the  new  was  connected  with  the  old  ring  or  not.  According  to  Mr. 
Bond's  measurements  the  breadth  of  the  new  ring  is  1".5. 

"  The  same  appearances  were  noticed  by  the  Rev.  W.  R.  Dawes,  at  his  ob- 
servatory, near  Maidstone,  in  England,  on  the  25th  and  29th  of  November,  and 
subsequently  by  Mr.  Lassell,  of  Starfield,  near  Liverpool." 


NOTES  xi,  xn.  375 

Mr.  G.  P.  Bond  has  propounded  a  bold  and  ingenious  theory  relative  to  the 
physical  constitution  of  Saturn's  rings ;  which  is,  that  "  they  are  in  a  fluid  state. 
and  within  certain  limits  change  their  form  and  position  in  obedience  to  the 
laws  of  equilibrium  of  rotating  bodies."  He  conceives,  also,  that  under  peculiar 
circumstances  of  disturbance  several  subdivisions  of  the  two  fluid  rings  may 
take  place,  and  continue  for  a  short  time  until  the  sources  of  disturbance  are 
removed,  when  the  parts  thrown  off  would  again  reunite.  He  supports  his 
theory  by  arguments  drawn  from  the  results  of  observation,  and  by  certain 
physical  considerations.  The  chief  argument  derived  from  observation  is,  that 
several  apparent  subdivisions  of  the  double  ring  have  been  noticed  by  different 
observers  from  time  to  time,  and  that  these  have  in  general  been  invisible  to  the 
same  observers  with  the  same  telescopes,  and  under  equally  favorable  circum- 
stances, and  have  also  entirely  escaped  the  observation  of  many  other  observers 
provided  with  equally  good  telescopes.  It  is  supposed  that  these  facts  admit 
of  explanation  only  on  the  hypothesis  that  the  ring  is  a  fluid  mass,  capable  of 
occasional  subdivision.  (See  Mr.  Bond's  original  paper  on  this  subject,  published 
in  Nos.  25  and  26  of  the  Astronomical  Journal.) 

Professor  Peirce,  of  Harvard  University,  has  followed  up  the  speculations  of 
Mr.  Bond,  by  undertaking  to  demonstrate,  from  purely  mechanical  considera- 
tions, that  Saturn's  ring  cannot  be  solid.  "  I  maintain,  unconditionally,"  says  he, 
"  that  there  is  no  conceivable  form  of  irregularity  and  no  combination  of  irregu- 
larities, consistent  with  an  actual  ring,  which  would  serve  to  retain  it  perma- 
nently about  the  primary  if  it  were  solid." 

He  is  led  by  his  investigations  to  the  curious  result,  that  Saturn's  ring  it 
sustained  in  a  position  of  stable  equilibrium  about  the  planet  solely  by  the 
attractive  power  of  his  satellites ;  and  that  "  no  planet  can  have  a  ring  unless  it 
is  surrounded  by  a  sufficient  number  of  properly  arranged  satellites."  (See 
Astronomical  Journal  for  June  16th,  1851.) 


NOTE    XI. 

The  seventh  satellite  of  Saturn,  in  the  order  of  distance  from  the  primary, 
was  discovered  by  the  Messrs.  Bond,  with  the  great  refractor  of  the  Cambridge 
Observatory,  on  the  16th  of  September,  1848;  and  observed  two  days  after- 
wards by  Mr.  Lassell,  at  Starfield,  near  Liverpool,  with  his  large  reflector.  In 
fact,  it  appears  to  have  been  distinctly  made  out  to  be  a  satellite  by  these  two 
observers  on  the  same  night,  viz.,  that  of  the  19th  of  September. 

"  The  orbit  of  the  new  satellite  serves  to  fill  up  a  large  chasm  before  existing 
between  the  6th  and  8th  satellites  (see  Table  VI).  It  is  fainter  than  either  of  the 
two  interior  satellites  discovered  by  Sir  William  Herschel.  Its  time  of  revolur 
tion  is  about  21.18  days,  the  semi-axis  of  its  orbit,  at  the  mean  distance  of 
Saturn,  214",  and  Messrs.  Bond  and  Lassell  have  concurred  in  giving  it  the 
name  of  Hyperion." 

The  periods  of  revolution,  and  the  mean  distances  of  the  satellites  of  Saturn 
from  their  primary,  together  with  the  mythological  names  proposed  for  them  by 
Sir  John  Herschel,  are  given  in  Table  VI. 


NOTE   XII. 

"  Two  of  the  satellites  of  Uranus  are  much  more  conspicuous  than  the  rest. 
and  their  periods  and  distances  from  the  planet  have  been  ascertained  with 
tolerable  certainty.  They  are  the  second  and  fourth  of  those  set  down  in  the 
synoptic  table  (Table  VI).  Of  the  remaining  four,  whose  existence,  though  an- 
nounced with  considerable  confidence  by  their  original  discoverer,  could  hardly 
be  regarded  as  fully  demonstrated,  two  only  have  been,  hitherto  re-observed ; 


376  NOTE  xm. 

viz.,  the  first  of  our  table,  interior  to  the  two  larger  ones,  by  the  independent 
observations  of  Mr.  Lassell,  and  M.  Otto  Struve,  and  the  third,  intermediate  be- 
tween the  larger  ones,  by  the  former  of  these  astronomers.  The  remaining  two, 
if  future  observation  should  satisfactorily  establish  their  real  existence,  will 
probably  be  found  to  revolve  in  orbits  exterior  to  all  these."  (Herschel's  Out- 
lines of  Astronomy,  Art.  551.) 

It  is  just  announced  (Nov.  28th,  1851),  that  Mr.  Lassell  has  discovered  two 
new  satellites  attending  upon  Uranus.  The  following  information  is  communi- 
cated with  respect  to  them :  "  They  are  interior  to  the  innermost  of  the  two 
bright  satellites  first  discovered  by  Sir  William  Herschel,  and  generally  known 
as  the  second  and  fourth.  It  would  appear  that  they  are  also  interior  to  Sir 
William's  first  satellite,  to  which  he  assigned  a  period  of  revolution  of  about  five 
days  and  twenty-one  hours,  but  which  satellite  I  have  as  yet  been  unable  to 
recognize.  I  first  saw  these  two  of  which  I  now  communicate  the  discovery,  on 
the  24th  of  last  month,  and  had  then  little  doubt  that  they  would  prove  satel- 
lites. I  obtained  further  observations  of  them  on  the  28th  and  30th  of  October, 
and  also  last  night  (Nov.  2d),  and  find  that  for  so  short  an  interval  the  observa- 
tions are  well  satisfied  by  a  period  of  revolution  of  almost  exactly  four  days  for 
the  outermost,  and  two  days  and  a  half  for  the  closest.  They  are  very  faint  ob- 
jects ;  certainly  not  half  the  brightness  of  the  two  conspicuous  ones  ;  but  all  the 
four  were  last  night  steadily  visible,  in  the  quieter  moments  of  the  air,  with  a 
magnifying  power  of  778  on  the  20  ft.  equatorial." 

This  discovery  would  seem  to  confirm  the  inference  drawn  by  Mr.  Dawes,  from 
a  discussion  of  the  observations  formerly  made  by  Lassell  and  Struve  upon  the 
nearest  satellite.  He  considers  these  observations  incompatible  with  each  other. 
"  While  Struve's  observations  indicate  a  period  of  three  days  and  twenty  hours, 
LasselFs  observations  indicate  a  period  of  only  two  days  and  two  hours.  He 
therefore  infers  that  there  must  be,  at  least,  two  satellites  interior  to  that  which 
Herschel  denominates  the  second."  He  also  considers  it  doubtful  whether  the 
other  satellite  discovered  by  Lassell  is  really  Herschel  s  third  satellite,  as  stated 
above. 

It  would  seem,  therefore,  that  at  least  two,  and  perhaps  three,  of  the  Her- 
echelian  satellites  have  been  seen  by  later  observers,  and  that  two  new  satellites 
have  probably  been  discovered  by  Lassell.  Accordingly  Uranus  has  certainly 
three  satellites,  and  probably  as  many  as  eight. 

Neptune. 

The  apparent  diameter  of  Neptune  is  nearly  3",  and  its  actual  diameter  is 
41,500  miles.  "  To  two  observers  it  has  afforded  strong  suspicion  of  being  sur- 
rounded with  a  ring  very  highly  inclined  ;  and  from  the  observations  of  Mr.  Las- 
eell,  M.  Otto  Struve,  and  Mr.  Bond,  it  appears  to  be  attended  certainly  by  one, 
and  very  probably  by  two  satellites,  though  the  existence  of  the  second  can  hardly 
yet  be  considered  as  quite  demonstrated."  (For  the  details  of  the  interesting 
history  of  the  discovery  of  this  planet,  see  Herschel's  Outlines  of  Astronomy,  or 
Loomis's  Progress  of  Astronomy.) 

THE  NEW  ASTEROIDS, 

Astrcea,  Hebe,  Iris,  Flora,  Metis,  Hygeia,  Parthenope,  Clio,  Egeria,  Irene, 

Eunomia. 

Of  the  dimensions  and  other  physical  peculiarities  of  these  planetary  bodies, 
no  knowledge  has  as  yet  been  obtained,  further  than  that  they  are  very  small 
bodies,  and  probably  inferior  in  size  to  the  other  four  asteroids.  They  are  all 
of  about  the  ninth  apparent  magnitude,  except  Metis,  which  is  of  the  tenth  or 
eleventh. 


NOTE   XIII. 

Certain  remarkable  phenomena  were  exhibited  by  Biela's  comet  at  its  hut 
return  (in  1846),  an  account  of  which  will  be  found  in  Note  V. 


NOTES  XIV,   XV.  37? 


NOTE    XIV. 

The  great  problem  of  the  determination  of  the  parallax  and  distance  of  a 
fixed  star,  first  solved  by  Bessel,  has  since  been  undertaken,  with  success  by 
other  astronomers.  The  following  is  a  list  of  the  most  reliable  determinations 
which  have  been  hitherto  obtained : 

a  Centauri 0".913  (Henderson). 

61  Cygni  0  .348  (Bessel). 

aLyrae 0  .261  (Struve). 

Sirius 0  .230  (Henderson). 

'  n   lnfij  Peters,  Struve,  Preuss, 

Polar18 ; °  -106i      andLindenau. 

In  the  case  of  the  Pole  Star,  the  estimated  error  to  which  the  result  obtained 
is  liable,  is  l/9  of  the  parallax.  For  the  other  stars  it  is  a  still  smaller  fraction. 
The  parallax  of  the  pole  star  indicates  a  distance  which  light  would  require 
more  than  80  years  to  traverse. 

The  measurements  for  a  Lyrse,  as  well  as  for  61  Cygni,  were  made  with  a 
micrometer.  Professor  Henderson  determined  the  parallax  of  a  Centauri,  from 
a  discussion  of  a  series  of  observations  upon  that  star  made  by  him,  with  a  large 
mural  circle,  in  the  years  1832  and  1838,  at  the  Royal  Observatory  of  the  Cape 
of  Good  Hope.  Subsequent  observations  with  a  similar  but  more  efficient  in- 
strument by  Mr.  Maclear,  have  conducted  to  very  nearly  the  same  result.  The 
observations  by  M.  Peters  were  made  with  the  great  vertical  circle  of  the  Pul- 
kova  Observatory.  His  observations  with  this  instrument  upon  61  Cygni  gave 
a  parallax  almost  identical  with  that  found  by  Bessel.  This  same  observer  has 
also  undertaken  to  determine  the  parallax  of  several  other  stars,  with  the  fol- 
lowing results:  Arcturus  (0".127),  Iota  Ursse  Majoris  (0".138),  1830  Groom- 
bridge  (0".226),  Capella  (0".046),  a  Cygni  (no  measurable  parallax).  But  the 
probable  errors  are  one-half,  or  more,  of  the  parallaxes  found. 


NOTE   XV. 

It  is  an  interesting  fact,  ascertained  by  M.  Argelander,  of  Bonn,  that  the 
periods  of  several  of  the  variable  stars  are  subject  to  a  slow  alteration.  The 
two  stars,  Omicron  Ceti  and  Algol,  may  be  cited  as  examples.  It  is  conjectured 
that  these  variations  of  period  are  periodical 

Sir  John  Herschel,  in  his  "  Outlines  of  Astronomy,"  gives  a  list  of  thirty -four 
variable  stars  whose  periods  have  been  approximately  or  roughly  determined, 
but  each  year  adds  to  the  number.  There  are  many  other  stars  known  to  be 
variable,  but  whose  periods  and  limits  of  variation  of  brightness  are  unknown. 

The  statement  made  in  the  text  of  the  second  general  fact  noticed  with 
respect  to  variable  stars  should  read  thus :  they  pass  from  their  epoch  of  least 
light  to  that  of  their  greatest  in  considerably  less  time  than  from  their  greatest 
to  their  least. 

"  The  alterations  of  brightness  in  the  southern  star  n  Argus,  which  have  been 
recorded,  are  very  singular  and  surprising.  In  the  time  of  Halley  (1677)  it 
appeared  as  a  star  of  the  fourth  magnitude.  Lacaille,  in  1751,  observed  it  of  the 
second;  in  the  interval  from  1811  to  1815  it  was  again  of  the  fourth;  and 
again,  from  1822  to  1826,  of  the  second.  On  the  1st  of  February,  1827,  it  was 
noticed  by  Mr.  Burchell  to  have  increased  to  the  first  magnitude,  and  to  equal 
a  Crucis.  Thence  again  it  receded  to  the  second  ;  and  so  continued  until  the  end 
of  1837.  All  at  once,  in  the  beginning  of  1838,  it  suddenly  increased  in  lustre 
so  as  to  surpass  all  the  stars  of  the  first  magnitude,  except  Sirius,  Canopus,  and 
a  Centauri,  which  last  star  it  nearly  equalled.  Thence  it  again  diminished,  but 
this  time  not  below  the  first  magnitude,  until  April,  1843,  when  it  had  again 
increased  so  as  to  surpass  Canopus,  and  nearly  equal  Sirius  in  splendor.  A 

48 


/ 
*v 


378  NOTES  xvi-xvni. 

strange  field  of  speculation  is  opened  by  this  phenomenon.  The  temporary 
stars  heretofore  recorded  have  all  become  totally  extinct.  Variable  stars,  so  far 
as  they  have  been  carefully  attended  to,  have  exhibited  periodical  alternations, 
in  some  degree,  at  least,  regular,  of  splendor  and  comparative  obscurity.  But 
here  we  have  a  star  fitfully  variable  to  an  astonishing  extent,  and  whose  fluctu- 
ations are  spread  over  centuries,  apparently  in  no  settled  period,  and  with  no 
regularity  of  progression.  "What  origin  can  we  ascribe  to  these  sudden  flashes 
and  relapses  ?  What  conclusions  are  we  to  draw  as  to  the  comfort  or  habita- 
bility  of  a  system  depending  for  its  supply  of  light  and  heat  on  so  uncertain  a 
source."  (Herschel's  Outlines.) 


NOTE   XVI. 

It  must  be  conceded  that  the  change  in  the  length  of  the  periods  of  the  vari- 
able stars,  noticed  in  the  previous  note,  is  apparently  at  variance  with  the 
theory  given  in  the  text,  since  all  analogy  teaches  that  the  periods  of  rotation 
should  be  uniform.  Argelander,  who  has  studied  the  phenomena  of  variable 
stars  more  attentively  than  any  other  observer,  is  of  the  opinion  that  "  the  time 
has  not  come  in  which  we  should  prepare  to  frame  a  theory.  The  minute 
changes  characterizing  the  phenomena  have  been  too  little  studied  and  dis- 
cussed." 


NOTE   XVII. 

"Among  the  most  remarkable  triple,  quadruple,  or  multiple  stars,  may  be 
enumerated, 


a  Andromedae. 
s  Lyrae. 
?;  Cancri. 


0  Orionis. 
H  Lupi. 
H  Bootis. 


£  Scorpii. 

11  Monocerotis. 

12  Lyncis. 


Of  these  a  Andromedae,  //  Bootis,  and  /a  Lupi,  appear  in  telescopes  even  of 
considerable  optical  power,  only  as  ordinary  double  stars ;  and  it  is  only  when 
excellent  instruments  are  used  that  their  smaller  companions  are  subdivided 
and  found  to  be  in  fact  extremely  close  double  stars,  t  Lyrae  offers  the  remark- 
able combination  of  a  double-double  star.  Viewed  with  a  telescope  of  low 
power,  it  appears  as  a  close  and  easily  divided  double  star  ;  but  on  increasing 
the  magnifying  power,  each  individual  is  perceived  to  be  beautifully  and  closely 
double,  the  one  pair  being  about  2£",  the  other  about  3"  asunder.  Each  of  the 
stars,  £  Cancri,  £  Scorpii,  11  Monocerotis,  and  12  Lyncis,  consists  of  a  principal 
star,  closely  double,  and  a  smaller  and  more  distant  attendant,  while  0  Orionis 
presents  the  phenomenon  of  four  brilliant  principal  stars,  of  the  respective  4th, 
6th,  7th,  and  8th  magnitudes,  forming  a  trapezium,  the  longest  diagonal  of  which 
is  21".4,  and  accompanied  by  two  excessively  minute  and  very  close  companions, 
to  perceive  both  of  which  is  one  of  the  severest  tests  which  can  be  applied  to  a 
telescope."  (Herschel's  Outlines.) 


NOTE  XVIII, 

Later  observations  have  led  to  the  discovery  that  the  star  «  Indi  has  a  greater 
motion  than  any  other  star, — the  amount  of  its  annual  displacement  being 


NOTE  XIX. 


379 


An  interesting  confirmation  of  the  solar  motion  mentioned  in  Art.  593  has 
recently  been  obtained  by  Mr.  Galloway,  from  a  discussion*  of  certain  observa- 
tions made  at  different  epochs  and  by  different  observers  upon  eighty-one  stars 
of  the  southern  hemisphere.  He  concludes  from  his  discussion,  that  the  point 
towards  which  the  sun's  motion  is  directed,  is  situated  in  R.  A.  260°  1'  and  N. 
Dec.  34°  23' ;  "  a  result  so  nearly  identical  with  that  afforded  by  the  northern 
hemisphere  as  to  afford  a  full  conviction  of  its  near  approach  to  truth,  and  what 
may  fairly  be  considered  a  demonstration  of  the  physical  cause  assigned." 


NOTE   XIX. 

The  following,  according  to  Herschel,  are  the  places,  for  1830,  of  the  principal 
globular  clusters,  as  specimens  of  their  class : — 


E.  A. 

N.  P.  D. 

B.A. 

N.  P.  D. 

B.A. 

N.  P.  D. 

h.  m.   i. 

o   / 

h.   m.  a. 

o   / 

h.   m.   s. 

o   / 

0  16  25 

163  2 

15  9  56 

87  16 

17  26  51 

143  34 

9  8  33 

154  10 

15  34  56 

127  13 

17  28  42 

93  8 

12  47  41 

159  57 

16  6  55 

112  33 

18  26  4 

114  2 

13  4  30 

70  55 

16  23  2 

102  40 

18  55  49 

150  14 

18  16  38 

136  35 

16  35  37 

53  13 

21  21  43 

78  34 

13  34  10 

60  46 

16  50  24 

119  61 

21  24  40 

91  34 

Many  of  the  nebulous  objects  in  the  heavens  hitherto  classed  among  resolvable 
nebulae,  have  lately  been  resolved  by  the  magnificent  reflecting  telescope  con- 
structed by  Lord  Rosse ;  and  many  nebulae  which  have  offered  no  appearance 
of  stars  to  all  previous  observers,  and  which  were  supposed  by  the  elder  Her- 
schel to  be  collections  of  nebulous  matter,  have  either  been  partially  resolved 
by  this  telescope,  or  have  assumed  in  it  the  appearance  of  resolv ability.  In 
view  of  these  facts  it  must  be  conceded,  that  "  although  nebulae  do  exist,  which 
even  in  this  powerful  telescope  appear  as  nebulae,  without  any  sign  of  resolu- 
tion, it  may  very  reasonably  be  doubted  whether  there  be  really  any  essential 
physical  distinction  between  nebulae  and  clusters  of  stars,  at  least  in  the  nature 
of  the  matter  of  which  they  consist,  and  whether  the  distinction  between  such 
nebulae  as  are  easily  resolved,  barely  resolvable  with  excellent  telescopes,  and 
altogether  irresolvable  with  the  best,  be  any  thing  else  than  one  of  degree, 
arising  merely  from  the  excessive  minuteness  and  multitude  of  the  stars,  of 
which  the  latter,  as  compared  with  the  former,  consist." 

Sir  James  South,  who  made  a  trial  of  Lord  Rosse's  monster  telescope  in 
March,  1845,  gives  the  following  account  of  his  observations: — "Never  before  hi 
my  life  did  I  see  such  glorious  sidereal  pictures  as  this  instrument  afforded  us. 
The  most  popularly  known  nebulae  observed  were  the  ring  nebula  in  the  Canes 
Venatici,  which  was  resolved  into  stars  with  a  magnifying  power  of  548,  and 
the  94th  of  Messier,  which  is  in  the  same  constellation,  and  which  was  resolved 
into  a  large  globular  cluster  of  stars,  not  much  unlike  the  well-known  cluster  in 
Hercules.  On  subsequent  nights  observations  of  other  nebulae,  amounting  to 
some  thirty  or  more,  removed  most  of  these  from  the  list  of  nebulae,  where  they 
had  long  figured,  to  that  of  clusters ;  while  some  of  these  latter  exhibited  a 
sidereal  picture  in  the  telescope  such  as  man  before  had  never  seen,  and  which, 
for  its  magnificence,  baffles  all  description." 

The  following  are  some  of  the  nebulae  which  have  assumed  a  new  and  re- 
markable appearance  when  viewed  through  Lorjl  Rosse's  telescopes,  of  3  ft.  and 
6  ft.  aperture : 

1.  The  Crab-nebula.  To  previous  observers  this  curious  object  presented  the 
appearance  of  an  oval  resolvable  nebula.  "  Lord  Rosse's  three  feet  reflector 
exhibits  it  with  resolvable  filaments  singularly  disposed,  springing  principally 
from  its  southern  extremity,  and  not,  as  is  usual,  in  clusters,  irregularly  in  all 


380  NOTE  XIX. 

directions.    It  is  studded  with  stars,  mixed,  however,  with  a  nebulosity,  probably 
consisting  of  stars  too  minute  to  be  recognized." 

2.  The  Dumb-bell  nebula,  so  named  from  its  resemblance  to  a  dumb-bell,  as 
shown  by  Sir  John  Herschel's  drawing  (see  Nichol's  Architecture  of  the  Heavens), 
in  Lord  Rosse's  8  ft,  telescope,  has  quite  a  different  appearance,  and  is  seen  to 
consist  of  innumerable  stars  mixed  with  nebulosity. 

3.  The  nebula  in  the  Dog's  Ear  was  formerly  described  as  having  the  form  of 
a  ring,  divided  through  about  one-third  of  its  course  into  two  separate  branches 
or  streams,  and  thus  regarded  as  presenting  a  singular  counterpart  to  our  own 
Milky  Way.     In  Lord  Rosse's  six  feet  reflector  "the  former  simple  shape  is 
transformed  into  a  scroll,  apparently  'unwinding  with  numerous  filaments,  and  a 
mottled  appearance,  which  looks  like  the  breaking  up  of  a  cluster."      It  has 
accordingly  received  the  designation  of  the  Scroll  or  Spiral  nebula. 

4.  The  great  nebula  in  Orion  has  also  been  divested  of  the  mystery  in  which 
it  has  so  long  remained  enshrouded,  by  the  same  telescope.      Lord  Rosse  says  : 
"  I  may  safely  say  that  there  can  be  little  if  any  doubt  as  to  the  resolvability  of 
this  nebula.    We  can  plainly  see  that  all  about  the  trapezium  is  a  mass  of  stars ; 
the  rest  of  the  nebula  also  abounding  with  stars,  and  exhibiting  the  characteris- 
tics of  resolvability  strongly  marked" 

Mr.  Bond,  with  the  great  refractor  at  Cambridge,  has  also  succeeded  in  resolv- 
ing the  brighter  portion  of  this  nebula  immediately  adjacent  to  the  trapezium, 
or  the  sextuple  star  8. 

The  great  nebula  in  Andromeda,  mentioned  in  the  text,  has  also  been  care- 
fully observed  with  the  Cambridge  refractor,  and  decisive  evidence  obtained  of 
its  resolvability. 

Detailed  descriptions  of  these  two  nebulae,  as  seen  with  the  Cambridge  tele- 
scope, accompanied  with  accurate  drawings,  have  been  published  by  the  Messrs. 
Bond  (Transactions  of  the  American  Academy  of  Arts  and  Sciences,  vol.  iii). 

In  the  southern  hemisphere  there  are  two  remarkable  nebulous  masses  of 
light,  conspicuously  visible  to  the  naked  eye,  which  are  known  by  the  name  of 
Magellanic  Clouds,  or  Nubeculce  (major  and  minor).  Sir  John  Herschel  de- 
scribes them  as  being  in  the  appearance  and  brightness  of  their  light  not  unlike 
portions  of  the  Milky  Way  of  the  same  apparent  size,  and  round  or  oval  in  their 
general  form. 

"  When  examined  through  powerful  telescopes,  the  constitution  of  the  nube- 
culae, and  especially  of  the  nubecula  major,  is  found  to  be  of  astonishing  com- 
plexity. The  general  ground  of  both  consists  of  large  tracts  and  patches  of 
nebulosity,  in  every  stage  of  resolution,  from  light  irresolvable  with  18  inches  of 
reflecting  aperture,  up  to  perfectly  separated  stars  like  the  Milky  Way,  and 
clustering  groups  sufficiently  insulated  and  condensed  to  come  under  the  desig- 
nation of  irregular,  and  in  some  cases  pretty  rich  clusters.  But,  besides  these, 
there  are  also  nebulae  in  abundance,  both  regular  and  irregular ;  globular  clus- 
ters in  every  state  of  condensation ;  and  objects  of  a  nebulous  character  quite 
peculiar,  and  which  have  no  analogue  in  any  other  region  of  the  heavens.  Such 
is  the  concentration  of  these  objects,  that  in  the  area  occupied  by  the  nubecula 
major,  not  fewer  than  278  nebulae  and  clusters  have  been  enumerated,  besides 
60  or  60  outliers,  which  (considering  the  general  barrenness  in  such  objects  of 
the  immediate  neighborhood)  ought  certainly  to  be  reckoned  as  its  appendages, 
being  about  6£  per  square  degree,  which  very  far  exceeds  the  average  of  any 
other,  even  the  most  crowded  parts  of  the  nebulous  heavens.  In  the  nubecula 
minor  the  concentration  of  such  objects  is  less,  though  still  very  striking,  37  hav- 
ing been  observed  within  its  area,  and  6  adjacent  but  outlying.  The  nubeculae, 
then,  combine,  each  within  its  own  area,  characters  which  in  the  rest  of  the 
heavens  are  no  less  strikingly  separated ;  viz.,  those  of  the  galactic  and  the 
nebular  system.  Globular  clusters  (except  in  one  region  of  small  extent)  and 
nebulae  of  regular  elliptic  forms  are  comparatively  rare  in  the  Milky  Way,  and 
are  found  congregated  in  the  greatest  abundance  in  a  part  of  the  heavens  the 
most  remote  possible  from  thai  circle ;  whereas,  in  the  nubeculae  they  are  indis- 
criminately mixed  with  the  general  starry  ground,  and  with  irregular  though 
email  nebulae."  (Uerschel's  Outlines  of  Astronomy.) 


NOTES   XX.    XXI.  381 


NOTE  XX. 

According  to  Struve,  the  nebula  in  Andromeda  is  1$°  long  by  16'  broad,  and 
thus  nearly  one-half  greater  than  the  moon's  disk.  Mr.  G.  P.  Bond  describes  it 
as  extending  nearly  2£°  in  length,  and  upwards  of  1°  in  breadth. 

Since,  as  stated  in  Note  XIX,  many  of  the  nebulae,  which  were  supposed  by 
Sir  William  Herschel  to  be  masses  of  nebulous  matter,  have  recently  been  found 
to  consist  of  stars,  it  must  now  be  regarded  as  exceedingly  doubtful  whether 
any  such  supposed  nebulous  masses  really  exist  in  space ;  and,  on  the  other 
hand,  highly  probable  that  all  the  irresolvable  nebulae  are  only  vast  beds  of 
stars  either  too  remote,  or  composed  of  too  small  or  too  closely  compacted  stars, 
to  appear  otherwise  than  one  general  mass  of  cloudy  light  in  the  best  tele- 
scopes. 


NOTE   XXI. 

Struve,  of  the  Pulkova  Observatory,  in  a  recent  work  entitled  6tudes  d' As- 
tronomic Stellaire,  has  undertaken  to  establish  that  the  stratum  of  the  Milky 
Way  is  really  fathomless  (at  least  in  every  direction  except,  perhaps,  at  right 
angles  to  the  stratum),  and  shows,  by  quotations  from  his  later  papers  en  the 
Milky  Way,  that  Sir  William  Herschel  was  led  finally  to  entertain  the*  same 
opinion,  in  opposition  to  the  views  he  had  at  first  expressed  (in  1785).  Accord- 
ingly, by  comparing  the  number  of  stars  seen  in  the  field  of  view  of  a  telescope 
when  pointed  in  two  different  directions  into  space,  we  do  not  obtain  the  rela- 
tive distances  through  to  the  boundaries  of  the  stratum  of  the  Milky  Way,  but 
only  the  relative  condensation  of  the  stars,  or  relative  density  of  the  starry 
stratum  in  the  two  directions.  Every  augmentation  in  the  power  of  the  tele- 
scope brings  into  view,  in  these  directions,  other  stars  before  invisible. 

Struve  remarks  :  "  It  may  be  asked  why  astronomers  have  generally  main- 
tained the  old  theory  concerning  the  Milky  Way,  propounded  in  1785,. although 
it  had  been  entirely  abandoned  by  the  author  himself,  as  we  have  demonstrated. 
I  believe  that  the  explanation  must  be  sought  in  two  circumstances.  It  was  a 
complete  system,  imposing  from  the  boldness  and  geometric  precision  of  its 
construction,  and  which  the  author  has  never  revoked  as  a  whole.  In  his  trea- 
tises, published  since  1802,  we  meet  with  only  partial  views,  but  which  are  suf- 
ficient, when  they  are  compared  together,  to  exhibit  the  final  idea  of  the  great 
astronomer." 

Sir  John  Herschel  does  not  give  his  assent  to  the  opinion  expressed  by  Struve. 
He  remarks : — "  Throughout  by  far  the  larger  portion  of  the  extent  of  the  Milky 
Way  in  both  hemispheres,  the  general  blackness  of  the  ground  of  the  heavens 
on  which  its  stars  are  projected,  and  the  absence  of  that  innumerable  multitude 
and  excessive  crowding  of  the  smallest  visible  magnitudes,  and  of  glare  pro- 
duced by  the  aggregate  light  of  multitudes  too  small  to  affect  the  eye  singly, 
which  the  contrary  supposition  would  seem  to  necessitate,  must,  we  think,  be 
considered  unequivocal  indications  that  its  dimensions  in  directions  where  these 
conditions  obtain,  are  not  only  not  infinite,  but  that  the  space-penetrating  power 
of  our  telescopes  suffices  fairly  to  pierce  through  and  beyond  it." 

If  it  be  true  that  the  stratum  of  the  Milky  Way  is  really  fathomless — that 
infinite  space  is  occupied  by  an  infinite  number  of  shining  stars,  the  central  suns 
of  planetary  systems  clustered  around  them,  ay  first  suggested  by  Kant,  then  it 
has  been  shown  by  Olbers  that  the  aspect  of  the  sky  should  be  that  of  a  vault 
shining  in  all  directions  with  a  lustre  similar  to  that  of  the  sun.  The  conclusion, 
therefore,  is  inevitable,  either  that  the  bed  of  stars  in  which  our  sun  is  posited 
is  not  infinite  in  extent,  or  that  space  is  not  perfectly  transparent ;  in  other 
words,  that  the  light  coming  from  the  stars  suffers  a  partial  extinction,  propor- 
tional in  amount  to  the  distance  traversed  by  it.  The  latter  view  was  advo- 
cated by  Olbers,  and  is  also  adopted  by  Struve,  who  by  means  of  this  concep- 
tion endeavors  to  reconcile  his  views  of  the  boundless  extent  of  our  firmament 


382  NOTE  xxn. 

with  the  feeble  luminosity  of  the  sky.  He  conceives,  upon  a  detailed  investiga 
tion,  that  the  actual  luminosity  of  the  sky  in  different  directions  is  adequately 
explained,  in  accordance  with  his  theory  of  the  unlimited  extent  of  the  stratum 
of  the  Milky  Way,  if  it  be  allowed  that  the  light  of  the  stars  suffers  an  extinc- 
tion of  only  Vioo  in  traversing  a  distance  equal  to  that  of  a  star  of  the  first  mag 
nitude.  Upon  this  supposition  the  extinction  for  the  most  distant  stars  visible 
in  telescopes  would  amount  to  88  per  cent. 

Herschel  urges,  in  opposition  to  this  theory,  that  "  if  applicable  to  any,  it  is 
equally  so  to  every  part  of  the  galaxy.  We  are  not  at  liberty  to  argue  that  at 
one  part  of  its  circumference  our  view  is  limited  by  this  sort  of  cosmical  veil 
which  extinguishes  the  smaller  magnitudes,  cuts  off  the  nebulous  light  of  distant 
masses,  and  closes  our  view  in  impenetrable  darkness ;  while  at  another  we  are 
compelled  by  the  clearest  evidence  telescopes  can  afford  to  believe  that  star- 
strewn  vistas  lie  open,  exhausting  their  powers  and  stretching  out  beyond  their 
utmost  reach,  as  is  proved  by  that  very  phenomenon  which  the  existence  of 
such  a  veil  would  render  impossible,  viz.,  infinite  increase  of  number  and  dimi- 
nution of  magnitude,  terminating  in  complete  irresolvable  nebulosity." 


NOTE   XXII. 

Or  Bather,  when  the  planets  are  compared  with  respect  to  density,  it  will  be 
seen  that  they  may  be  divided  into  two  classes,  viz. :  one  class,  comprising  Mer- 
cury, Venus,  the  Earth,  and  Mars,  each  of  which  has  a  density  nearly  equal  to 
unity  ;  and  a  second  class,  consisting  of  Jupiter,  Saturn,  Uranus,  and  Neptune, 
whose  density  is  between  0.13  and  0.23. 

It  is  a  curious  fact  that  the  same  classification  holds  with  respect  to  magni- 
tude and  period  of  rotation. 


* :  «* 


TABLE  I. 


Latitudes  and  Longitudes  from  the  Meridian  of  Greenwich,  of 
some  citiest  and  other  conspicuous  places. 


Names  of  Places. 

Latitude. 

Longitude 
in  Degrees. 

Longitude 
in  Time. 

Albany,  Capitol,                   New  York, 
Altona,  Obs.,                         Denmark, 
Baltimore,  Bait.  Mon't,    }    Maryland, 
Berlin,  Obs  ,                           Germany, 
Boston,  State  House,            Massach'ts, 

0      /      // 

42  39  3  N 
53  32  45  N 
39  17  23  N 
52  31  13  N 
42  21  28  N 

o     /    // 
73  44  49  W 
9  56  89  E 
76  37  30  W 
13  23  52  E 
71    4    9W 

4  54  59.3 
0  39  46  6 
5    6  30 
0  53  a5  5 
4  44  16.6 

Bremen,  Obs.,                       Germany, 
Brunswick,  Eowdoin  Coll.,   Maine, 
Canton,                                   China, 
Cape  of  Good  Hope,  Obs.,   Africa, 
Cape  Horn,                           S.  America, 

53  436  N 
43  53  ON 
23  8  9N 
33  56  3S 
55  58  41  S 

8  48  58E 
69  55    1  W 
113  16  54  E 
18  23  45  E 
67  1053  W 

0  35  15.9 
4  39  40 
7  33    8 
1  13  o5.0 
4  23  43 

Charleston,  St.  Mich'*  Ch.,  S.  Carolina,- 
Charlottesville,  Univers.,      Virginia, 
Cincinnati,  Fort  Wash  ,       Ohio, 
Copenhagen,  Obs.,                Denmark, 
Dorpat,  Obs.,                        Russia, 

32  46  33N 
38  2  3N 
39  5  54N 
55  40  53  N 

53  22  47  N 

79  57  27  W 
78  31  29  W 
84  27    0  W 
12  34  57  E 
26  43  45E 

5  19  49  8 
5  14    6 
5  37  43 
0  50  19  8 
1  4655 

Dublin,  Obs.,                        Ireland, 
Edinburgh,  Obs.,                   Scotland, 
Gptha,  Obs.,                          Germany, 
Gottingen.  Obs.,                    Germany, 
Greenwich,  Obs.,                 England, 

53  23  13  N 
55  57  23  N 

50  56  5N 
51  31  48  N 
51  28  39  N 

6  20  SOW 
3  10  54  W 
10  44    6  E 
9  5637E 
000 

0  25  22 
0  12  43  6 
0  42  56.4 
0  39  46.5 
000 

Konigsberg,  Obs.,                  Prussia, 
London,  St.  Paul's  Ch.,       England, 
Marseilles,  Obs.,                   France, 
Milan,  Obs.,                           Italy, 
Naples,  Obs.,                        Italy, 

54  42  SON 
51  30  49  N 
43  17  50  N 
4528  IN 
40  51  47  N 

20  30    7E 
0    5  48W 
5  22  15  E 
9  11  48  E 
14  Id    4  E 

1  22    *5 

0    0  23 
0  21  29  0 
0  36  47.2 
057    0.3 

New  Haven.  College,            Connecticut, 
New  Orleans,  City  Hall,       Louisiana, 
New  York,  City  Hall,          New  York, 
Palermo,  Obs.,                     Italy, 
Paramatta,  Obs.,                  New  Holl'd, 

41  18  30  N 
29  57  45  N 
40  42  40N 
38  6  44N 

33  48  SOS 

72  56  45  W 
90    6  49  W 
74    1    8W 
13  21  24  E 
151    1  34  E 

4  51  47 
6    027 
4  56    45 
0  5325.6 
10    4    6.3 

Paris,  Obs.,                           France, 
Petersburgh,  Obs.,                Russia, 
Philadelphia,  IncTce  Hall,     Pennsylv'a, 
Point  Venus,                          Otaheite, 
Princeton,  College,               New  Jersey, 

4850  13  N 
59  56  31  N 
39  56  59  N 
17  29  21  S 
40  20  41  N 

2  20  24  E 

30  18  57  E 
75    9  54  W 
149  23  55  W 
74  39  33  W 

0    921.6 
2    1  15.8 
5    0  39.6 
957  56 
4  58  38.2 

Providence,  University,        Rhode  Isl'd, 
Quebec,  Castle,                    L.  Canada, 
Richmond,  Capitol,             Virginia, 
Rome,  Roman  College,         Italy, 
Savannah,  Exchange,           Georgia, 

41  49  22  N 
46  49  12  N 
37  32  17  N 
41  53  52  N 
32  456N 

71  24  48  W 
71  16    0  W 
77  27  2SW 
12  28  40  E 
81    8  18  W 

4  45  39  2 
4  45    4 
5    »50 
0  49  54.7 
52433 

Schenectady,                        New  York, 
Stockholm,  Obs.,                  Sweden, 
Turin,  Obs.,                           Italy, 
Vienna,  Obs.,                       Austria, 
Wardhus,                              Lapland, 
"Washington,  Capitol,           Diet.  Colum. 

4248  N 
59  2031  N 
45  4  6N 
48  1235  N 
70  22  36  N 
38  53  84  N 

73  55       W 
18    3  44  E 
7  42    6E 
1623    OE 
31    7  54  E 
77    1  30  \V 

4  55  40 
1  12  15 
0  30  48.4 
1    5  32 
2    4  32 
586 

2  TABLE  II.     Elements  of  the,  Planetary  Orbits. 

Epoch  for  Vesta,  Juno,  Ceres,  and  Pallas,  July  23d,  1831,  mean  noon  at 
Berlin  :  for  the  other  planets,  Jan.  1, 1801,  mean  noon  at  Greenwich.* 


Planet's      Inclination  to     q^    v        Longitude  of  As-     q.-  v»r        Longitude  of 
Name.        the  Ecliptic.      Sec.  Var.     cending  Node.      bec' Var'         Perihelion. 


Mercury 

Venus 

Earth 

Mars 

Vesta 

Juno 

Ceres 

Pallas 

Jupiter 

Saturn 

Uranus 


7    0    9.1 
3  23  285 

1  51    62 

7  7  57.3 
13  2  10  0 
10  36  55  7 
34  35  49.1 

1  18  51.3 

2  29  35.7 
0  46  28.4 


+  18.2 

—  46 

—  0.2 

—  12 

—  44 

—  22.6 

—  15.5 
+    3.1 


45  57  30  9 
74  51  55 

48  0  35 
103  20  28.0 
170  52  34.5 

80  53  49.7 
172  38  29  8 

98  26  18  9 
111  56  37.4 

72  59  35.3 


70.44 
51.10 

4167 
26 


+  25 

57.18 
,    51.12 

+  23.58 


± 


74  21  46  9 
128  43  53  1 

99  31  9.9 
332  23  56  6 
249  11  37. 

54  17  12.7 
147  41  23.5 
121  5  0.5 

11    8  34.6 

89  9  29.8 
167  31  16.1 


Sec.  Var. 


+  9322 
+  78.30 
-j-  103.15 

109.71 

157 


+  202 

+  9459 
+  115.68 
+  87.44 


Ptonot'a  ivrnm«.  Mean  Distance  from  Mean  Distance  from  Eccentricity  in  Parts 
flanet  s  JMame.   gun.  or  Semi-axis.          Sun  in  Miles.  of  the  Semi-axis. 


Sec.  Variation. 


Mercury 

Venus 

Earth 

Mars 

Vesta 

Juno 

Ceres 

Pallas 

Jupiter 

Saturn 

Uranus 


03S70981 
0.7233316 
1.0000000 
1.5236923 
2.3614800 
2.6694600 
2.7709100 
2.7726300 
5.2027760 
9.5387861 
191823900 


36814000 

68787000 

95103000 

144908000 

224584000 

253874000 

263522000 

263685000 

494797000 

907162000 

1824290000 


020551494 
000686074 
001678357 
0.09330700 
0.08856000 
0.25556000 
0.07673780 
0.24199800 
004816210 
0.05615050 
0.04661080 


+  .000003866 

—  .000062711 

—  .000041630 
+  .000090176 
+  .000004009 

—  .000005830 

+  .000159350 

—  .000312402 

—  .000025072 


'Janet's  Name. 


Mean  Longitude  at 
the  Epoch. 


Mean  Sidereal  Period 
Mean  Solar  Days. 


n  Motion  in  mean  Lon. 
in  1  yr.  of  365  days. 


Mercury 

Venus 

Earth 

Mars 

Vesta 

Juno 

Ceres 

Pallas 

Jupiter 

Saturn 

Uranus 


166    .0  48.6 

11  33    30 

100  39  13  3 

64  22  55.5 

84  47    3,2 

74  39  43  6 

307    3  25.6 

290  38  11.8 

112  15  23.0 

135  20    6.5 

177  48  23.0 


87.9692580 

224.7007869 

365.2563770 

6869796458 

1325.4850000 

1593  0670000 

16847350000 

1686.3050000 

4332.5848212 

10759.2198174 

30686.8208296 


53  43  3.6 
224  47  29.7 
—0  14  19.5 
191  17  9.1 


30  20  31.9 

12  13  36.1 

4  17  45.1 


Mean  Daily  Motion 
in  Longitude. 


4  5  32.6 
1  36  7.8 
0  59  8.3 
0  31  26.7 
0  16  17.9 
0  13  33.7 
0  12  49.4 
0  12  48.7 
0  4  59.3 
0  2  0.6 
0  0  42.4 


TABLE  III.— Elements  of  Moon's  Orbit.    Epoch,  Jan.  1, 1801. 


Mean  inclination  of  orbit      ... 
Mean  longitude  of  node  at  epoch 
Mean  longitude  of  perigee  at  epoch     - 
Mean  longitude  of  moon  at  epoch 
Mean  distance  from  earth,  or  semi-axis 
Eccentricity  in  parts  of  serai-axis 

Mean  sidereal  revolution  - 

Mean  tropical       do.         ... 

Mean  synodical    do. .... 

Mean  anomalistic  do.  - 

Mean  nodical        do.  - 

Mean  revolution  of  nodes ;  aider. 

Mean  revolution  of  perigee  ;  sider.    - 


d     A  m 


5    8  47.9 
13  53  17.7 
266  10    7.5 
118  17    8.3 
59r.  964350 
0.0548442 

27    7  43  11.5  =  27.32166142 
27    7  43    4.7  =  27  32158242 
29  12  44    2.9  =  29.53058872 
27  13  18  37.4  =  27.55459950 
27    5    5  36.0  =  27.21222222 
:  6793d.279 ;      trop.  =  6798d.l7707 
i ;  trop.  =  3231d.4751 


Element*  of  Meptune.—  Mean  distance,  30.0368000;  Period,  60126-V7100000;  Eccentricity, 
0  0087195 :  Inclination  of  orbit,  1°  46'  55T.O ;  Long,  of  Node,  130°  5'  ll".0 ;  Long,  of  Perihelion, 
470  IV  5G".7;  M.Long.  at  Epoch,  330°  44'  41".8 ;  Epoch,  1848,  Jan.  1,  Oh.  G.  T. 

*  For  an  accurate  table  of  the  Elements  of  all  the  Asteroids,  see  Note  HI. 


TABLE  IV.  3 

Diameters,  Volumes,  Masses,  dec,,  of  Sun,  Moon,  and  Planets. 


Apparent  Diameter. 

Equatorial 
Diameter.* 

Equatorial 
Diameter, 
in  Mites.* 

Volume. 

Least. 

At  Mean 
Distance. 

Greatest. 

Mercury 

5.0 

it 

6.5 

12.0 

0.396 

3140 

0.062 

Venus 

9.6 

16.5 

61.2 

0.984 

7800 

0.952 

Earth 

1.000 

7926 

1.000 

Mars 

3.6 

5.8 

18.3 

0.517 

4100 

0.138 

Jupiter 

30.0 

36.9 

45.9 

10.976 

87000 

1233.412 

Saturn 

16.2 

9.987 

79160 

900.000 

Uranus 

3,9 

4.353 

34500 

82.759 

Neptune 

/       // 

3.0 

/      n 

/       // 

5.236 

4J500 

144.008 

Sun 

31  31.0 

32  1.8 

32  35.6 

112.020 

887870 

1410366.376 

Moon 

29  21.9 

31  7.0 

33  31.1  |       0.273 

2163 

0.020 

Mass.f 

Density4 

Gravity. 

Sidereal  Rotation4 

Ltir- 

Mercury 

TraSTST 

1.12 

0.47 

7i.    m.    s. 
24     5  28.3 

6.680 

Venus 

nmrjv 

0.92 

0.93 

23  21  21.9 

1.911 

Earth 

TTsfjTT 

1.00 

1.00 

33  56     4.1 

1.000 

Mars 

_     I  _ 

0.95 

0.50 

24  37  20.4 

.431 

Jupiter 

HTIT^TT 

0.24 

2.85 

9  55  26.6 

.037 

Saturn 

sTffT-tfTTtf 

0.14 

1.03 

10  29  16.8 

.011 

Uranus 

_    l 

0.24 

0.76 

.003 

Neptune 

T™ 

0.14 

0.69 

.001 

Sun 

1 

0.25 

28.65 

607  48 

Moon 

rtnWw 

0.57 

0.15 

27     7  43 

TABLE  V. 
Elements  of  the  Retrograde  Motion  of  the  Planets. 


Planets. 

Mercury 
Venus 
Mars 
Jupiter 
Saturn 
Uranus 

Are  of 
Retrogradation. 

Duration  of 
Retrogradation. 

Elongation  at  the 
Stations. 

Synodic 
Revolution. 

0                       0        ' 

9  22  to  15  44 
14  35  to  17  12 
10    6  to  19  35 
9  51  to    9  59 
6  41  to    6  55 
336 

d      k            d      h 
23  12  to    21  12 
40  21  to    43  12 
60  18  to    80  15 
116  18  to  122  12 
138  18  to  135    9 
151 

o       '             o     " 
14  49  to    20  51 
27  40  to    29  41 
128  44  to  146  37 
113  35  to  11642 
107  25  to  110  46 
103  30 

days 
116 
584 
780 
399 
378 
370 

Satellites  of  Neptune. — "  One  only  has  certainly  been  observed — its  approximate 
period  being  5d.  20h.  50m.  45s. ;  distance  about  12  radii  of  the  planet." 

*  According  to  Herschel,  except  the  diameters  of  the  Sun  and  Moon. 

f  According  to  Encke,  with  the  exception  of  the  mass  of  Neptune,  which  ia  Professor  Peirce'a 
determination  from  Bond's  and  Lassel's  observations  of  the  satellite.  By  Leverrier's  second 
determination  the  mass  of  Mercury  is  siro^innr* 

$  According  to  Hansen  and  Midler,  in  the  case  of  the  planets. 


TABLE  VI. 


Elements  of  the  Orbits  of  the  Satellites. 

The  distances  are  expressed  in  equatorial  radii  of  the  primaries, 
periods  are  expressed  in  mean  solar  days. 

I.  Satellites  of  Jupiter. 


Th« 


Sat. 

Mean  Distance, 

Sidereal 
Revolution. 

Inclination  of 
Orbit  to  that 
of  Jupiter. 

Epoch 
of  Ele- 
ments. 

Mass  ;  that  of 
Jupiter  being 
1,000,000,000. 

1 
2 
3 
4 

6.04853 
9.62347 
15.35024 
26.99835 

d      h     m      s 

1  18  27  33.506 
3  13  14  36.393 
7    3  42  33.362 
16  16  31  49.702 

O        '          " 

3     5    30 

Variable. 
Variable. 
2    58  48 

Jan.  1, 

1801. 

17328 
23235 

88497 
42659 

II.   Satellites  of  Saturn. 


Name  and 
Order  of 
Satellite. 

Mean 
Distance. 

Sidereal 
Revolution. 

M,  Long,  at  the 
Epoch. 

Eccentricity 
and  Perisatur- 
nium. 

Epoch 
of  Ele- 
ments. 

d     A      m      s 

O            '           " 

1.  Mimas 

3.3607 

0  22  37  22.9 

256    58    48 

1790.0 

2.  Enceladus 

4.3125 

1     8  53     6.7 

67    41     36 

1836.0 

3.  Tethys 

5.3396 

1  21  18  25.7 

313     43    48 

0.04(?)—  54(?) 

Ditto 

4.  Dione 

6.8398 

2  17  41     8.9 

327    40    48 

0.02(?)—  42(8) 

Ditto 

5.  Rhea 

9.5528 

4  12  25  10.8 

353     44      0 

0.02(?)—  95(?) 

Ditto 

6.  Titan 

22.1450 

15  22  41  255 

137    21     24 

)  .029314 
f  256°  38' 

1830.0 

7.  Hyperion 

28.± 

22  12   ? 

8.  lapetus 

64.3590 

79     7  53  40.4 

269    37    48 

1790.0 

The  longitudes  are  reckoned  in  the  plane  of  the  ring  from  its  descending  node  with 
the  ecliptic.  The  first  seven  satellites  move  in  or  very  nearly  in  its  plane ;  that  of  the 
8th  lies  about  half-way  between  the  planes  of  the  ring  and  of  the  planet's  orbit.  The 
apsides  of  Titan  have  a  direct  motion  of  30'  28"  per  annum  in  longitude  (on  the  ecliptic). 

III.  Satellites  of  Uranus. 


Sat. 

Mean 
Distance. 

Sidereal 
Revolution. 

Epochs  of  Passage 
through  Ascending 
Node  of  Orbits.  Q.  T. 

Inclination  to  Ecliptic. 

d     h     m     s 

The  orbits  are  inclined  at 

1 
2 
3 

4 

17.0 

19.8  (?) 
22.8 

4(0 
8  16  56  31.3 
10  23(?) 
13  11     7  12.6 

1787,  Feb.  1  6th,  0  10 
1787,  Jan.  7th,   0  28 

an  angle  of  about  78°  58' 
to  the  ecliptic  in  a  plane 
whose  ascending  node  is 
in  long.  165°  30'  (Equinox 

5 
6 

45.5  (?) 
91.0 

38    2(?) 
107  12(?) 

of  1798).     Their  motion 
is  retrograde.    The  orbits 
are  nearly  circular. 

TABLE  VII.   Saturn  s  Ring. 


Exterior  diameter  of  exterior  ring 176,418  miles. 

Interior  ditto 155,272  " 

Exterior  diameter  of  interior  ring 151,690  " 

Interior  ditto 117,339  " 

Equatorial  diameter  of  the  body 79,160  " 

Interval  between  the  planet  and  interior  ring 19,090  " 

Interval  of  the  ring* -. 1,791  " 

Thickness  of  the  rings  not  exceeding 250 

Ditto,  according  to  Professor  Bond,  not  exceeding 50  " 


Mean  Astronomical  Refractions. 
Barometer  30  in.     Thermometer,  Fah.  50°. 


Ap.Alt. 

Refr. 

Ap.  Alt. 

Refr. 

Ap.  Alt. 

Refr. 

Alt. 

Refr. 

0°  0' 

33'  51" 

4°  0' 

11'  52" 

12°  0' 

4'  28.1" 

42° 

l'.4.6" 

5 

32  53 

10 

11  30 

10 

4  24.4 

43 

1  2.4 

10 

31  58 

20 

11  10 

20 

4  20.8 

44 

1  0.3 

15 

31  5 

30 

10  50 

30 

4  17.3 

45 

0.58.1 

20 

30  13 

40 

10  32 

40 

4  13.9 

46 

56.1 

25 

29  24 

50 

10  15 

50 

4  10.7 

47 

54.2 

30 

28  37 

5  0 

9  58 

13  0 

4  7.5 

48 

52.3 

35 

27  51 

10 

9  42 

10 

4  4.4 

49 

50.5 

40 

27  6 

20 

9  27 

20 

4  1.4 

50 

48.8 

45 

26  24 

30 

9  11 

30 

3  58.4 

51 

47.1 

50 

25  43 

40 

8  58 

40 

3  55.5 

52 

45.4 

55 

25  3 

50 

.  8  45 

50 

3  52.6 

53 

43.8 

1  0 

24  25 

6  0 

8  32 

14  0 

3  49.9 

54 

42.2 

5 

23  48 

10 

8  20 

10 

3  47.1 

55 

40.8 

10 

23  13 

20 

8  9 

20 

3  44.4 

56 

39.3 

15 

22  40 

30 

7  58 

30 

3  41.8 

57 

37.8 

20 

22  8 

40 

7  47 

40 

3  39.2 

58 

36.4 

25 

21  37 

50 

7  37 

50 

3  36.7 

59 

35.0 

30 

21  7 

7  0 

7  27 

15  0 

3  34.3 

60 

33.6 

35 

20  38 

10 

7  17 

35  30 

3  27.3 

61 

32.3 

40 

20  10 

20 

7  8 

16  0 

3  20.6 

62 

31.0 

N  45 

19  43 

30 

6  59 

16  30 

3  14.4 

63 

29.7 

50 

19  17 

40 

6  51 

17  0 

3  8.5 

64 

28.4 

55 

18  52 

50 

6  43 

17  30 

3  2.9 

65 

27.2 

2  0 

18  29 

8  0 

6  35 

18  0 

2  57.6 

66 

25.9 

5 

18  5 

10 

6  28 

19 

2  47.7 

67 

24.7 

10 

17  43 

20 

6  21 

20 

2  38.7 

68 

23.5 

15 

17  21 

30 

6  14 

21 

2  30.5 

69 

22.4 

20 

17  0 

40 

6  7 

22 

2  23.2 

70 

21.2 

25 

16  40 

50 

6  0 

23 

2  16.5 

71 

19.9 

30 

16  21 

9  0 

5  54 

24 

2  10.1 

72 

18.8 

35 

16  2 

10 

5  47 

25 

2  4.2 

73 

17.7 

40 

15  43 

20 

5  41 

26 

1  58.8 

74 

16.6 

45 

15  25 

30 

5  36 

27 

53.8 

75 

15.5 

50 

15  8 

40 

5  30 

28 

49.1 

76 

14.4 

55 

14  51 

50 

5  25 

29 

44.7 

77 

13.4 

3  0 

14  35 

10  0 

5  ^0 

30 

40.5 

78 

12.3 

5 

14  19 

10 

5  15 

31 

36.6 

79 

11.2 

10 

14  4 

20 

5  10 

32 

33.0 

80 

10.2 

15 

13  50 

30 

5  5 

33 

29.5 

81 

9.2 

20 

13  35 

40 

5  (/ 

34 

26.1 

82 

8.2 

25 

13  21 

50 

4  56 

35 

23.0 

83 

7.1 

30 

13  7 

11  0 

4  51 

36 

20.0 

84 

6.1 

35 

12  53 

10 

4  47 

37 

17.1 

85 

5.1 

40 

12  41 

20 

4  43 

38 

14.4 

86 

4.1 

45 

12  28 

30 

4  39 

39 

11.8 

87 

3.1 

50 

12  16 

40 

4  35 

40 

9.3 

88 

2.0 

if 

12  3  I 

50 

4  31 

41 

6.9 

89 

1.0 

TABLE  IX 


Corrections  of  Mean  Refractions. 


Ap.Alt. 

[diffo 
-4-1  B 

dif.  fo 
—  1°F 

1  Ap.Alt. 

Dif.  for 
4-1  B. 

Dif.  for 
—  1°F 

Ap.  Alt. 

Dif.  fo 
+  IB. 

Dif.  for 
—  1°F 

Alt. 

Dif.  for 
+1B. 

Dif.  for 

-  -1°  F. 

O  ' 

0  0 

74 

8.1 

4  0 

24.1 

1.70 

12  0 

9.00 

0.556 

42 

2.16 

0.130 

5 

71 

7.6 

10 

23.4 

1.64 

10 

8.86 

.548 

43 

2.09 

.125 

10 

69 

7.3 

20 

22.7 

1.58 

20 

8.74 

.541 

44 

2.02 

.120 

15 

67 

7.0 

30 

22.0 

1.53 

30 

8.63 

.533 

45 

1.95 

.116 

20 

65 

6.7 

40 

21.3 

1.48 

40 

8.51 

.524 

46 

1.88 

.112 

25 

63 

6.4 

50 

20.7 

1.43 

50 

8.41 

.517 

47 

1.81 

.108 

30 

61 

6.1 

5  0 

20.1 

1.38 

13  0 

8.30 

.509 

48 

1.75 

.104 

35 

59 

5.9 

10 

19.6 

1.34 

10 

8.20 

.503 

49 

1.69 

.101 

40 

58 

5.6 

20 

19.1 

1.30 

20 

8.10 

.496 

50 

1.63 

.097 

45 

56 

5.4 

30 

18.6 

1.26 

30 

8.00 

.490 

51 

1.58 

.094 

50 

55 

5.1 

40 

18.1 

1.22 

40 

7.89 

.482 

52 

1.52 

.090 

55 

53 

4.9 

50 

17.6 

1.19 

50 

7.79 

.476 

53 

1.47 

.088 

1  0 

52 

4.7 

6  0 

17.2 

1.15 

14  0 

7.70 

.469 

54 

1.41 

.085 

5 

50 

4.6 

10 

16.8 

1.11 

10 

7.61 

.464 

55 

1.36 

.082 

10 

49 

4.5 

20 

16.4 

1.09 

20 

7.52 

.458 

56 

1.31 

.079 

15 

48 

4.4 

30 

16.0 

1.06 

30 

7.43 

.453 

57 

1.26 

.076 

20 

46 

4.2 

40 

15.7 

1.03 

40 

7.34 

.448 

58 

1.22 

.073 

25 

45 

4.0 

50 

15.3 

1.00 

50 

7.26 

.444 

59 

1.17 

.070 

30 

44 

3.9 

7  0 

150 

0.98 

15  0 

7.18 

.439 

60 

1.12 

.067 

35 

43 

3.8 

10 

14.6 

.95 

15  30 

6.95 

.424 

61 

1.08 

.065 

40 

42 

3.6 

20 

14.3 

.93 

16  0 

6.73 

.411 

62 

1.04 

.062 

45 

40 

3.5 

30 

14.1 

.91 

16  30 

6.51 

.399 

63 

.99 

.060 

50 

39 

3.4 

40 

13.8 

.89 

17  0 

6.31 

.386 

64 

.95 

.057 

55 

39 

3.3 

50 

13.5 

.87 

17  30 

6.12 

.374 

65 

.91 

.055 

2  0 

38 

3.2 

8  0 

13.3 

.85 

18  0 

5.94 

.362 

66 

.87 

.052 

5 

37 

3.1 

10 

13.1 

.83 

19 

5.61 

.340 

67 

.83 

.050 

10 

36 

3.0 

20 

12.8 

.82 

20 

5.31 

.322 

68 

.79 

.047 

15 

36 

2.9 

30 

12.6 

.80 

21 

5.04 

.305 

69 

.75 

.045 

20 

35 

2.8 

40 

12.3 

.79 

22 

4.79 

.290 

70 

.71 

.043 

25 

34 

2.8 

50 

12.1 

.77 

23 

4.57 

.276 

71 

.67 

.040 

30 

33 

2.7 

9  0 

11.9 

.76 

24 

4.35 

.284 

72 

.63 

.038 

35 

33 

2.7 

10 

11.7 

.74 

25 

4.16 

.252 

73 

.59 

.036 

40 

32 

2.6 

20 

11.5 

.73 

26 

3.97 

.241 

74 

.56 

033 

45 

32 

2.5 

30 

11.3 

.72 

27 

3.81 

.230 

75 

.52 

.031 

50 

31 

2.4 

40 

11.1 

.71 

28 

3.65 

.219 

76 

.48 

.029 

55 

30 

2.3 

50 

11.0 

.70 

29 

3.50 

.209 

77 

45 

.027 

3  0 

30 

2.3 

10  0 

10.8 

.69 

30 

3.36 

.201 

78 

.41 

.025 

5 

29 

2.2 

10 

10.6 

.67 

31 

3.23 

.193 

79 

.38 

.023 

10 

29 

2.2 

20 

10.4 

.65 

32 

3.11 

.186 

80 

.34 

.021 

15 

28 

2.1 

30 

10.2 

.64 

33 

2.99 

.179 

81 

.31 

.018 

20 

28 

2.1 

40 

10.1 

.63 

34 

2.88 

.173 

82 

.27 

.016 

25 

•27 

2.0 

50 

9.9 

.62 

35 

2.78 

.167 

83 

.24 

.014 

30 

27 

2.0 

11  0 

9.8 

.60 

36 

2.68 

.161 

84 

.20 

.012 

35 

26 

2.0 

10 

9.6 

.59 

37 

2.58 

.155 

85 

.17 

.010 

40 

26 

1.9 

20 

9.5 

.58 

38 

2.49 

.149 

86 

.14 

.008 

45 

25 

1.9 

30 

9.4 

.57 

39 

2.40 

.144 

87 

.10 

.006 

50 

25 

1.9 

40 

9.2 

.56 

40 

2.32 

.139 

88 

.07 

.004 

55 

25 

1.8 

50 

9.1 

.55 

41 

2.24 

.134 

89 

.03 

.002 

TABLE  X. 


Parallax  of  the  Sun,  on  the  first  day  of  each  Month:  the  mean 
horizontal  Parallax  being  assumed  =  8  ",60. 


Alti- 
tude. 

Jam 

Feb. 
Dec. 

March. 

Nov. 

April. 
Oct. 

May. 
Sept. 

June. 
Aug. 

July. 

o 
0 

8.75 

8.73 

8.67 

8.60 

8.53 

8.48 

8.46 

5 

8.73 

8.69 

8.64 

8.56 

8.50 

8.44 

8.42 

10 

8.62 

8.59 

8.54 

8.47 

8.40 

8.35 

8.33 

15 

8.45 

8.43 

8.38 

8.30 

8.24 

8.19 

8.17 

20 

8.22 

8.20 

8.15 

8.08 

8.01 

7.97 

7.95 

25 

7.93 

7.91 

7.86 

7.79 

7.73 

7.68 

7.67 

30 

7.58 

7.56 

7.51 

7.45 

7.39 

7.34 

7.33 

35 

7.17 

7.15 

7.11 

7.04 

6.99 

6.94 

6.93 

40 

6.70 

6.68 

6.64 

6.59 

-&S3 

6.49 

6.48 

45 

6.19 

6.17 

6.13 

6.08 

6.03 

5.99 

5.98 

50 

5.62 

5.61 

5.58 

5.53 

5.48 

5.45 

5.44 

55 

5.02 

5.01 

4.98 

4.93 

4.89 

4.86 

4.85 

60 

4.37 

4.36 

4.34 

4.30 

4.26 

4.24 

4.23 

65 

3.70 

3.69 

3.67 

3.63 

3.60 

3.58 

3.57 

70 

2.99 

2.98 

2.97 

2.94 

2.92 

2.90 

2.89 

75 

2.26 

2.26 

2.25 

2.23 

2.21 

2.19 

2.19 

SO 

1.52 

1.52 

1.51 

1.49 

1.48 

1.47 

1.47 

85 

0.76 

0.76 

0.76 

0.75 

0.74 

0.74 

0.74 

90 

0.00 

0.00 

0.00 

0.00 

000 

0.00 

0.00 

TABLE  XL 

Semi-diurnal  Arcs* 


Declination. 

Lat. 

1° 

5° 

10° 

15° 

20° 

25o 

30o 

o 

h  m 

h  in 

h  m 

A  m 

h  m 

h  m 

h  m 

5 

6  0 

6  2 

6  4 

6  5 

6  7 

6  9 

6  12 

10 

6  1 

6  4 

6  7 

6  11 

6  15 

6  19 

6  24 

15 

6  1 

6  5 

6  11 

6  16 

6  22 

6  29 

6  36 

20 

6  1 

6  7 

6  15 

6  22 

6  30 

6  39 

6  49 

25 

6  2 

6  9 

6  19 

6  29 

6  39 

6  50 

7  2 

30 

6  2 

6  12 

6  23 

6  36 

6  49 

7  2 

7  18 

35 

6  3 

6  14 

6  28 

6  43 

6  59 

7  16 

7  35 

40 

6  3 

6  17 

6  34 

6  52 

7  11 

7  32 

7  56 

45 

6  4 

6  20 

6  41 

7  2 

7  25 

7  51 

8  21 

50 

6  5 

6  24 

6  49 

7  14 

7  43 

8  15 

8  r,4 

55 

6  6 

6  29 

6  58 

7  30 

•8  5 

8  47 

9  42 

60 

6  7 

6  35 

7  11 

7  51 

8  36 

9  35 

12  0 

65 

6  9 

6  43 

7  29 

8  20 

9  25 

12  0 

TABLE   XII. 

Equation  of  Time,  to  convert  Apparent  Time  into  Mean  Time 
Argument,  Mean  Longitude  of  the  Sun. 


0* 

I* 

II* 

III* 

IV* 

V« 

0 

min.  sec. 

min.  sec. 

min.  sec. 

min.  sec. 

min.  sec. 

min.  sec. 

0 

+  6  58.4 

—  1  29.7 

•—338.7 

•f  1  27.0 

+  6    4.1 

-f  2  49.7 

1 

639.7 

142.0 

334.2 

140.1 

6    6.3 

2  34.5 

2 

620.9 

1  53.8 

329.1 

1  53.1 

6    8.0 

218.9 

3 

6    2.1 

2    5,2 

323.5 

2    6.0 

6    9.1 

2    2.8 

4 

543.3 

2  15.9 

3  17.3 

2  18.9 

6    9.5 

1  46.4 

5 

524.5 

226.1 

3  10.7 

231.7 

6    9.3 

1  29.5 

6 

5    5.7 

235.9 

3    3.5 

244.3 

6    8.5 

1  12.3 

7 

446.9 

245.0 

2  56.0 

256.7 

6    7.2 

054.6 

8 

428.2 

253.6 

247.9 

3    8.9 

6    5.2 

036.6 

9 

4    9.6 

3    1.8 

239.5 

320.8 

6    2.5 

-f-0  18.2 

10 

351.1 

3    9.3 

230.5 

332.5 

559.3 

—  0    0.4 

11 

332.6 

3  16.3 

221.2 

343.9 

555.4 

019.5 

12. 

3  14.3 

322.8 

211.5 

355.0 

551.0 

038.8 

13 

2  56.2 

328.6 

2    1.4 

4    5.8 

545.8 

058.4 

14 

238.3 

333.9 

1  51.0 

416.3 

540.1 

1  18.2 

15 

220.5 

338.6 

140.1 

4  26.5 

533.7 

1  38.3 

16 

2    3.0 

342.7 

1  29.0 

436.3 

5  26.7 

1  58.5 

17 

145.7 

346.3 

1  17.6 

445.7 

519.2 

2  19.1 

18 

1  28.6 

349.2 

1    5.9 

454.7 

5  11.1 

239.8 

19 

I  11.7 

351.5 

054.1 

5    3.3 

5    2.3 

3    0.7 

20 

055.2 

353.3 

042.0 

511.3 

453.0 

321.6 

21 

039.1 

354.4 

029.6 

5  18.9 

443.1 

342.8 

22 

023.3 

355.0 

0  17.1 

526.0 

432.7 

4    4.0 

23 

+  0    7.8 

3  55.0 

—  0    4.4 

532.6 

421.6 

425.3 

24 

—  0    7.3 

3  54.5 

-1-0    8.4 

538.6 

4  10.1 

446.6 

25 

022.0 

353.3 

021.5 

544.2 

357.9 

5    8.1 

26 

036.3 

351.5 

034.5 

5  49.3 

345.3 

529.5 

27 

050.3 

349.2 

047.6 

553.9 

332.1 

551.0 

28 

1    3.8 

346.2 

1    0.7 

557.8 

3  18.5 

6  12.3 

29 

1  16.9 

342.8 

1  13.8 

6    1.2 

3    4.3 

6  33.7 

30 

—  1  29.7 

—  338.7 

+  1  27.0 

+  6    4.1 

+  2  49.7 

—  654.9 

TABLE  XIII. 

Secular   Variation  of  Equation  of  Time. 
Argument,  Sun's  Mean  Longitude. 


0* 

I* 

II* 

III* 

IV* 

V* 

sec. 

sec. 

sec. 

sec. 

sec. 

sec. 

sec. 

0 

—  3 

+  4 

+  11 

+  14 

f  13 

+  9 

3 

2 

5 

11 

14 

13 

8 

6 

1 

6 

12 

14 

12 

8 

9 

—  1 

6 

12 

15 

12 

7 

12 

0 

7 

12 

14 

12 

W   i 

15 

+  1 

8 

13 

14 

11 

I 

18 

2 

8 

13 

14 

11 

6 

21 

2 

9 

14 

14 

10 

5 

24 

3 

9 

14 

14 

10 

5 

27 

4 

10 

14 

14 

9 

4 

30 

+  4 

-f  11 

+  14 

-r  13 

+  9 

+  4 

TABLE   XII 


Equation  of  Time,  to  convert  Apparent  Time  into  Mean  Time. 
Argument,  Mean  Longitude  of  the  Sun. 


VI* 

VII* 

VIII* 

IX* 

X* 

XI* 

o 

min.  sec. 

min.  sec. 

min.  sec. 

min.  sec. 

min.  sec. 

min.  see. 

0 

—    654.9 

—  15  18.9 

—  13  58.7 

—   1  30.6 

+  1130.0 

+  14    3.1 

1 

7  16.1 

15  27.9 

13  43.0 

1    0.2 

1147.0 

13  56.0 

2 

737.2 

1536.1 

13  26.3 

—  029.8 

12    3.3 

13  48.4 

3 

758.3 

15  43.7 

13    8.9 

+    0    0.6- 

12  18.7 

1340.1 

4 

8  19.1 

15  50.5 

12  50.5 

031.0 

12  33.4 

1331.1 

5 

839.8 

15  56.5 

1231.4 

1    1.3 

12  47.2 

13  21.6 

6 

9    0.2 

16    1.8 

1211.6 

131.4 

13    0.1 

13  11.4 

7 

920.5 

16    6.3 

1151.1 

v    2    1.3 

13  12.2 

13    0.7 

8 

940.6 

16    9.9 

11  29.9 

231.0 

13  23.5 

12  49.4 

9 

10    0.3 

16  12.9 

11    7.9 

3    0.5 

13  33.9 

12  37.4 

10 

10  19.8 

16  15.1 

10  45.4 

329.7 

13  43.6 

12  25.0 

11 

10  38.9 

16  16.5 

10  22.0 

358.6 

13  52.3 

12  12.2 

12 

10  57.8 

16  17.0 

958.1 

427.1 

14    0.2 

11  58.9 

13 

IT  16.2 

16  16.6 

933.5 

455.2 

14    7.3 

11  45.1 

14 

11  34.4 

16  15.4 

9    8.4 

5  22.9 

14  13.5 

11  30.9 

15 

1152.1 

16  13.4 

8  42.6 

550.2 

14  18.9 

11  16.3 

16 

12    9.5 

16  10.4 

816.4 

6  17.1 

14  23.4 

11    1.1  X 

17 

12  26.5 

16    6.7 

749.6 

643.5 

14  27.2 

10  45.6 

18 

12  42.9 

16    2.1 

722.5 

7    9.3 

14  30.0 

10  29.7 

19 

12  58.9 

15  56.6 

6  54.9 

734.6 

14  32.  1 

10  13.5 

20 

13  14.4 

1550.1 

627.0 

759.3 

14  33.3 

956.9 

21 

13  29.5 

15  42.9 

558.5 

823.4 

14  33.7 

940.1 

22 

1344.1 

15  34.8 

529.7 

"8  46.9 

14  33.3 

923.0 

23 

13  58.0 

15  25.8 

5    0.5 

9    9.8 

14  32.2 

9    5.7    ; 

24 

1411.4 

15  16.0 

431.0 

932.0 

14  30.2 

848.0    j 

25 

1424.1 

15    5.2 

4    1.4 

9  53.5 

14  27.5 

830.2 

26 

14  36.3 

14  53.6 

331.6 

10  14.3 

14  24.0 

8  12.2 

27 

14  47.9 

1441.1 

3    1.5 

"  10  34.4 

14  19.9 

754.0 

28 

14  58.8 

14  27.7 

231.3 

10  53.8 

14  15.0 

735.5 

29 

15    9.2 

14  13.6 

2    1.0 

11  12.3 

14    9.4 

7  17.0 

30 

—  15  18.9 

—  1358.7 

—    1  30.6 

+  1130.0 

+  14    3.1 

+    658.4 

TABLE  XIII. 

Secular  Variation  of  Equation  of  Time. 
Argument,  Sun's  Mean  Longitude. 


VI* 

VII* 

VIII* 

IX* 

X* 

XI* 

o 

sec. 

sec. 

sec. 

sec. 

sec. 

sec. 

0 

+4 

—  2 

—10 

—15 

—15 

—10 

3 

3 

3 

10 

15 

14 

10 

6 

3 

4 

11 

15 

14 

9 

9 

2 

4 

12 

15 

14 

8 

12 

1 

5 

12 

15 

13 

8 

15 

+  1 

6 

13 

15 

13 

7 

18 

0 

7 

13 

15 

12 

6 

21 

0 

7 

14 

15 

12 

5 

24 

—1 

8 

14 

15 

11 

5 

27 

2 

9 

15 

15 

11 

4 

30 

—  2 

—10 

—15 

—15  |  —10 

—  9 

10 


TABLE  XIV. 


Perturbations  of  Equation  of  Time. 


III. 


II. 

0 

100 

200  300 

400 

50C 

600 

700 

800 

900 

1000 

sec. 

sec. 

sec. 

sec. 

sec. 

sec. 

sec. 

sec. 

sec. 

tec. 

sec. 

0 

1.4 

0.8 

.0 

1.7 

1.7 

1.2 

0.7 

0.4 

0.6 

1.4 

1.4 

100 

1.2 

1.4 

.1 

1.0 

1.6 

1.8 

1.1 

0.7 

0.6 

0.7 

1.2 

300 

0.9 

1.0 

.2 

1.2 

1.2 

1.5 

1.7 

1.1 

0.5 

0.7 

0.9 

300 

0.7 

1.1 

.1 

0.9 

1.2 

1.4 

1.5 

1.6 

1.2 

0.5 

0.7 

400 

0.5 

0.6 

.2 

1.2 

0.8 

1.0 

1.6 

1.7 

1.5 

1.2 

0.5 

500 

1.0 

0.5 

0.6 

1.2 

1.4 

0.8 

0.8 

1.5 

1.9 

1.5 

1.0 

600 

L7 

1.0 

0.4 

0.5 

1.2 

1.4 

0.9 

0.6 

1.3 

2.0 

1.7 

700 

1.9 

1.8 

.1 

0.4 

0.4 

1.1 

1.6 

1.1 

0.7 

1.2 

1.9 

800 

1.2 

1.8 

.8 

1.2 

0.4 

0.3 

1.0 

1.6 

1.2 

0.7 

1.2 

900 

0.7 

1.1 

.7 

1.8 

1.2 

0.6 

0.2 

0.8 

1.6 

1.3 

0.7 

1000 

1.4 

0.8 

1.0 

1.7 

1.7 

1.2 

0.7 

0.4 

0.6 

1.4 

1.4 

II. 

rv. 

0 

sec. 
0.6 

sec. 
0.7 

sec. 
0.5 

sec. 
0.3 

sec. 
0.2 

sec. 
0.6 

sec. 
0.7 

sec. 
0.5 

sec. 
0.2 

sec. 
0.1 

sec. 
0.6 

100 

0.2 

0.7 

0.6 

0.5 

0.2 

0.3 

0.6 

0.9 

0.5 

0.2 

0.2 

200 

0.2 

0.4 

0.6 

0.5 

0.4 

0.3 

0.4 

0.6 

0.5 

0.5 

0.2 

300 

0.4 

0.2 

0.5 

0.5 

0.5 

0.4 

0.4 

0.4 

0.5 

0.5 

0.4 

400 

0.5 

0.4 

0.4 

0.4 

0.4 

0.4 

0.5 

0.5 

0.4 

0.4 

0.5 

500 

0.4 

0.5 

0.5 

0.5 

0.4 

0.4 

0.3 

0.4 

0.5 

0.3 

0.4 

600 

0.3 

0.3 

0.5 

0.6 

0.4 

0.4 

0.3 

0.5 

0.7 

0.4 

0.3 

700 

0.4 

0.2 

0.3 

0.6 

•  0.6 

0.4 

0.2 

0.2 

0.7 

0.7 

0.4 

800 

0.6 

0.3 

0.2" 

0.3 

0.7 

0.6 

0.3 

02 

0.3 

0.8 

0.6 

900 

0.8 

0.5 

0.3 

0.1 

0.4 

0.7 

0.5 

0.3 

0.1 

0.5 

0.8 

1000 

0.6 

0.7 

0.5 

0.3 

0.2 

0.6 

0.7 

0.5 

0.2 

0.1 

0.6 

II. 

V, 

0 

sec. 
1.0 

sec. 
1.0 

sec. 
1.1 

sec. 
1.2 

sec. 
1.1 

sec. 
1.0 

sec. 
0.7 

sec. 
0.4 

sec. 
0.6 

sec. 
0.9 

sec. 
1.0 

100 

0.9 

0.9 

0.8 

1.0 

1.3 

1.3 

1.0 

0.7 

0.4 

0.5 

0.9 

200 

0.5 

0.7 

0.7 

0.8 

1.0 

1.0 

1.1 

1.2 

0.9 

0.3 

0.5 

300 

0.2 

0.5 

0.7 

0.7 

0.8 

1.2 

1.5 

1.5 

1.1 

0.5 

0.2 

400 

0.3 

0.2 

0.5 

0.7 

0.7 

0.9 

1.3 

1.4 

1.4 

1.0 

0.3 

500 

0.8 

0.3 

0.2 

0.5 

0.7 

0.7 

1.0 

1.4 

1.4 

1.4 

0.8 

600 

1.3 

0.7 

0.3 

0.3 

0.5 

0.7 

0.9 

1.1 

1.4 

1.6 

1.3 

700 

1.5 

1.1 

0.7 

0.3 

0.4 

0.5 

0.8 

1.0 

1.2 

1.4 

1.5 

800 

1.3 

1.3 

1.0 

0.7 

0.4 

0.4 

0.6 

0.8 

1.0 

1.2 

1.3 

900 

1.1 

1.2 

1.2 

1.0 

0.8 

0.6 

0.5 

0.6 

0.9 

1.1 

1.1 

1000 

1.0 

1.0 

1.1 

1.2 

1.1 

1.0 

0.7 

0.4 

0.6 

0.9 

1.0 

Moon  and  Nutation. 

I. 

tec. 
0.5 

sec. 
0.8 

sec.  5  sec. 
1.0  1.0 

sec. 
08 

sec. 
0.5 

*ec. 
0.2 

sec. 
0.0 

sec. 
0.0 

sec.  ]  sec. 
0.2  0.5 

N. 

0.1 

0.1  0.2  0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.1  10.1 

Constant  3».0 


TABLE  XV. 


11 


For  converting  any  given  day  into  the  decimal  part  of  a  year 
of  365  days. 


Day 

Jan. 

Feb. 

March 

April 

May 

June 

1 

.000 

.085 

.162 

.247 

.329 

.414 

2 

.003 

.088 

.164 

.249 

.331 

.416 

3 

.006 

.090 

.167 

.252 

.334 

.419 

4 

.008 

.093 

.170 

.255 

.337 

.422 

5 

.011 

.096 

.173 

.258 

.340 

.425 

6 

.014 

099 

.175 

.260 

.342 

.427 

7 

.016 

.101 

.178 

.263 

.345 

.430 

8 

.019 

.104 

.181 

.266 

.348 

.433 

9 

.022 

.107 

.184 

.268 

.351 

.436 

10 

.025 

.110 

.186 

.271 

.353  * 

.438 

11 

.027 

.112 

.189 

274 

.356 

.441 

12 

.030 

.115 

.192 

.277 

.359 

.444 

13 

.033 

.118 

.195 

.279 

.362 

.446 

14 

.036 

.121 

.197 

.282 

.364 

.449 

15 

.038 

.123 

.200 

.285 

.367 

.452 

16 

.041 

.126 

.203 

.288 

.370 

.455 

17 

.044 

.129 

.205 

.290 

.373 

.458 

18 

•046 

.132 

.208 

.293 

.375 

.460 

19 

.049 

.134 

.211 

.296 

.378 

.463 

20 

.052 

.137 

.214 

.299 

.381 

.466 

21 

.055 

.140 

.216 

.301 

.384 

.468 

22 

.058 

.142 

.219 

.304   , 

.386 

.471 

23 

.060 

.145 

.222 

.30? 

.389 

.474 

24 

.063" 

.148 

.225 

.310 

.392 

.477 

25 

.066 

.151 

.227 

.312 

.395 

.479 

26 

.068 

.153 

.230 

.315 

.397 

.483 

27 

.071 

.156 

.233 

.318 

.400 

.485 

28 

.074 

.159 

.236 

.321 

.403 

.488 

29 

.077 

.238 

.323 

.405 

.490 

30 

.079 

.241 

.326 

.408 

.493 

31 

.082 

.244 

.411 

12 


TABLE  XV.,  Continued. 


For  converting  any  given  day  into  the  decimal  part  of  a  year 
of  365  days. 


Day 

July 

August 

Sept. 

Oct. 

Nov. 

Dec. 

1 

.496 

.581 

.666 

.748 

.833 

.915 

2 

.499 

.584 

.668 

.751 

.836 

.918 

3 

.501 

.586 

671 

.753 

.838 

.921 

4 

.504 

.589 

.674 

.756 

.841 

.923 

5 

.507 

.592 

.677 

.759 

.844 

.926 

6 

.510 

.595 

.679 

.762 

846 

.929 

7 

.512 

.597 

.682 

.764 

.849 

.931 

8 

.515 

.600 

.685 

.767 

.852 

.934 

9 

.518 

.603 

.688 

.770 

855 

.937 

10 

.521 

.605 

.690 

.773 

.858 

.940 

11 

.'523 

.608 

.693 

.775 

.860 

.942 

12 

526 

611 

.696 

.778 

.863 

.945 

13 

.529 

614 

.699 

.781 

.866 

.948 

14 

.532 

.616 

.701 

.784 

.868 

.951 

15 

.534 

.619 

.704 

.786 

.871 

.953 

16 

.537 

.622 

.707 

.789 

.874 

.956 

17 

.540 

.625 

.710 

.792 

877 

.959 

18 

.542 

.627 

.712 

.795 

879 

.962 

19 

.545 

.630 

.715 

.797 

.882 

.964 

20 

.548 

.633 

.718 

.800 

885 

.967 

21 

.551 

.636 

.721 

.803 

888 

.970 

22 

.553 

.638 

.723 

805 

890 

.973 

23 

.556 

.641 

.726 

.808 

893 

.975 

24 

.559 

.644 

.729 

.811 

896 

.978 

25 

.562 

.647 

.731 

.814 

.899 

.981 

26 

.564 

.649 

.734 

.816 

.901 

.984 

27 

.567 

.652 

.737 

.819 

.904 

.986 

28 

.570 

.655 

.740 

.822 

.907 

.989 

29 

.573 

.658 

.742 

.825 

.910 

.992 

30 

.575 

.660 

.745 

.827 

.912 

.995 

31 

.578 

.663 

.830 

.997 

TABLE  XVI. 


13 


For  converting  time  into  decimal  parts  of  a  day. 


Hours 

Minutes 

Seconds 

h. 

m.        m. 

s.          s. 

1 

.04167 

1 

.00069 

31 

.02153 

1 

.00001  D  31 

.00036 

2 

.08333 

2 

.00139 

32 

.02222 

2 

.00002  1  32 

.00037 

3 

.12500 

3 

.00208 

33 

.02292 

3 

.00003  I  33 

.00038 

4 

.16667 

4 

.00278 

34 

.02301 

4 

.00005  |  34 

.00039 

5 

.20833 

5 

.00347 

35 

.02430 

5 

.00006 

35 

.00040 

6 

.25000 

6 

.00417 

36 

.02500 

6 

.00007 

36 

.00042 

7 

.29167 

7 

.00486 

37 

.02569 

7 

.00008 

37 

.00043 

8 

.33333 

8 

.00556 

38 

.02639 

8 

.00009 

38 

.00044 

9 

.37500 

9   .00625 

39 

.02708 

9 

.00010 

39 

.00045 

10 

.41667 

10 

.00694 

40 

.02778 

10 

.0001  a 

\  40 

.00046 

11 

.45833 

11 

.00764 

41 

.02847 

11 

.00013 

41 

.00047 

12 

.50000 

12 

.00833 

42 

.02917 

12 

.00014 

42 

.00049 

13 

.54167 

13  !  .00903 

43 

.02986 

13 

.00015 

43 

.00050 

14 

.58333 

14   .00972 

44 

.03056 

14 

.00016 

44 

.00051 

15 

.62500 

15 

01042 

45 

.03125 

15 

.00017 

45 

.00052 

16 

.66667 

16 

.01111 

46 

.03194 

16 

.00018 

46 

.00053 

17 

.70833 

17 

.01180 

47 

.03264 

17 

.00020 

47 

.00054 

18 

.75000 

18 

.01250 

48 

.03333 

18 

.00021 

48 

.00056 

19 

.79167 

19 

.01319 

49 

.03403 

19 

.00022 

49 

.00057 

20 

.83333 

20 

.01389 

50 

.03472 

20 

.00023 

50 

.00058 

21 

.87500 

21 

01458 

51 

.03542 

21 

.00024 

51 

.00059 

22 

.91667 

22 

.01528 

52 

.03611 

22 

.00025 

52 

.00060 

23 

.95833 

23 

01597 

53 

.03680 

23 

.00027 

53 

.00061 

24 

1.00000 

24 

.01667 

54 

.03750 

24 

.00028 

54 

.00062 

25 

.01736 

55 

.03819 

25 

.00029 

55 

.00064 

26 

.01805 

56 

.03889  ' 

26 

.00030 

56 

.00065 

27 

.01875 

57 

.03958 

27 

.00031 

57 

.00066 

28 

.01944 

58 

.04028 

28 

.00032 

58 

.00067 

29 

.02014 

59 

.04097 

29 

.00034 

59 

.00068 

30 

.02083 

60 

.04167 

30 

.00035 

60 

.00069 

14 


TABLE  XVII. 


For  converting  Minutes  and  Seconds  of  a  degree,  into  the 
decimal  division  of  the  same. 


Minutes 

Seconds 

1 

.OJ667 

31 

.51667 

// 
1 

.00028 

31 

.00861 

2 

.03333 

32 

.53333 

2 

.00056 

32 

.00889 

3 

.05000 

33 

.55000 

3 

.00083 

33 

.00917 

4 

.06667 

34 

.6*6667 

4 

.00111 

34 

.00944 

5 

.08333 

35 

.58333 

5 

.00139 

35 

.00972 

6 

.10000 

36 

.60000 

6 

.00167 

36 

.01000 

7 

.11667 

37 

.61667 

7 

.00194 

37 

.01028 

8 

.13333 

38 

.63333 

8 

.00222 

38 

.01056 

9 

.15000 

39 

.65000 

9 

.00250 

39 

.01083 

10 

.16667 

40 

.66667  - 

10 

.00278 

40 

.01111 

11 

.18333 

41 

.68333 

11 

.00306 

41 

.01139 

12 

.20000 

42 

.70000 

12 

.00333 

42 

.01167 

13 

.21667 

43 

.71667 

13 

.00361 

43 

.01194 

14 

.23333 

44 

.73333 

14 

.00389 

44 

.01222 

15 

.25000 

45 

.75000 

15 

.00417 

45 

.01250 

16 

.26667 

46 

.76667 

16 

.00444 

46 

.01278 

17 

.28333 

47 

.78333 

17 

.00472 

47 

.01306 

18 

.30000 

48 

.80000 

18 

.00500 

48 

.01333 

19 

.31667 

49 

.81667 

19 

.00528 

49 

.01361 

20 

.33333 

50 

.83333 

20 

.00556 

50 

.01389 

21 

.35000 

51 

.85000 

21 

.00583 

51 

.01417 

22 

.36667 

52 

.86667 

22 

.00611 

52 

.01444 

23 

.38333 

53 

.88333 

23 

.00639 

53 

.01472 

24 

.40000 

54 

.90000 

24 

.00667 

54 

.01500 

25 

.41667 

55 

.91667 

25 

.00694 

55 

.01528 

26 

.43333 

56 

.93333 

26 

.00722 

56 

.01556 

27 

.45000 

57 

.95000 

27 

.00750 

57 

.01583 

28 

.46667 

58 

.96667 

28 

.00778 

58 

.01611 

29 

.48333 

59 

.98333 

29 

.00806 

59 

.01639 

30 

.50000 

60 

1.00000 

30 

.00833 

60 

.01667 

TABLE  XVIII. 
Sun's  Epochs. 


15 


Years. 

M.  Long. 

Long.Peri. 

I 

II 

III 

IV 

V 

N 

VI 

VII 

a  o  /  " 

a  o  /  " 

1830 

9  10  37  46.9 

9  10  0  54 

228 

279 

169 

598 

758 

519 

989 

362! 

1831 

9  10  23  27.4 

9  10  1  55 

588 

278 

793 

130 

842 

573 

235 

396  j 

1832B. 

9  10  9  7.9 

9  10  2  57  I  948 

278 

418 

661 

926 

627 

482 

430  . 

1833 

9  10  53  56.8 

910  359 

342 

280 

47 

194 

11 

681 

764 

464 

1834 

9  10  39  37.3 

910  5  0 

702 

279 

671 

725 

95 

734 

11 

498 

1835 

9  10  25  17.8 

910  6  2 

62- 

279 

296 

256 

179 

788 

257 

532 

1836B. 

9  10  10  58.4 

910  7  3 

422 

278 

920 

788 

264 

842 

504 

566 

1837 

9  10  55  47.2 

910  8  5 

816 

280 

549 

321 

348 

895 

787 

600 

1838 

9  10  41  27.8 

9  10  9  6 

176 

279 

173 

852 

432 

949 

33 

634 

1839 

91027  8.3 

9  10  10  8 

536 

279 

798 

383 

511 

3 

279 

668 

1840B. 

9  10  1248.8 

91011  9 

896 

278 

422 

915 

601 

56 

526 

702 

1841 

9  10  57  37.7 

9  10  12  11 

290 

280 

51 

447 

685 

110 

809 

736 

1842 

9  10  43  18.2 

9  10  13  12 

650 

279 

676 

979 

770 

164 

55 

770 

1843 

9  10  28  58.8 

9  10  14  14 

10 

279 

300 

510 

854 

218 

301 

804 

1844B. 

9  10  14  39.3 

9  10  15  15 

370 

278 

924 

41 

938 

272 

548 

838 

1845 

9  10  59  28.2 

9  10  16  17 

764 

280 

553 

574 

23 

325 

831 

872 

1846 

9  10  45  8.7 

9  10  17  19 

124 

280 

177 

106 

107 

379 

77 

906 

1847 

9  10  30  49.2 

9  10  18  20 

484 

279 

802 

637 

191 

433 

324 

940 

1848B. 

9  10  *6  29.8 

9  10  19  22 

844 

278 

427 

168 

276 

487 

570 

974 

1849 

911  118.6 

9  10  20  23 

238 

280 

55 

700 

360 

540 

853 

8 

1850 

9  10  46  59.2 

9  10  21  25 

598 

280 

680 

231 

444 

594 

99 

41 

1851 

9  10  32  39.7 

9  10  22  26 

958 

279 

304 

762 

529 

648 

346 

75 

1852B 

9  10  18  20.2 

9  10  23  28 

319 

278 

929 

294 

613 

701 

592 

109 

1853 

911  3  9.1 

9  10  24  29 

713 

280 

557 

827 

697 

755 

875 

143 

1854 

9  10  48  49.6 

9  10  25  31 

73 

280 

182 

358 

782 

809 

121 

177 

1855 

9  10  34  30.2 

9  10  26  32 

133 

279 

806 

889 

866 

863 

368 

211 

1856B 

9  10  20  10.7 

9  10  27  34 

793 

279 

430 

421 

950 

916 

614 

245 

1857 

911  459.6 

9  10  28  35 

187 

281 

60 

953 

35 

970 

897 

279 

-1858 

9  105040.1 

9  10  29  37 

547 

280 

684 

485 

119 

24 

144 

313 

t!859 

9  10  36  20.7 

9  10  30  39 

907 

279 

308 

16 

203 

78 

390 

347 

1860B 

9  10  22  1.2 

9  10  31  40 

267 

279 

933 

547 

288 

131 

636 

381 

1861 

911  650.1 

9  10  32  42 

661 

281 

562 

80 

372 

185 

919 

415 

1862 

9  10  52  30.6 

9  10  33  43 

21 

280 

186 

612 

456 

239 

166 

449 

1863 

9  1038  11.1 

9  10  34  45 

381 

280 

810 

143 

541 

292 

412 

483 

1864B 

9  10  23  51.7 

9  10  35  46 

741 

279 

435 

674 

625 

346 

659 

517 

1865 

9118  40.5 

9  10  36  48 

135 

281 

64 

207 

709 

400 

941 

551 

1866 

9  105421.1 

9  10  37  49 

495 

280 

688 

738 

794 

453 

188 

585 

1867 

9  10  40  1.6 

9  10  38  51 

855 

280 

313 

270 

878 

507 

434 

619 

1868B 

9  10  25  42.2 

9  10  39  52 

215 

279 

937 

801 

962 

56 

681 

653 

1969 

9  11  1031.0 

9  10  40  54 

609 

281 

566 

334 

47 

615 

963 

687 

1870 

9  1050  11.6 

9  10  41  56 

969 

280 

190 

865 

131 

668 

210 

721 

16 


TABLE  XIX. 

Sun's  Motions  for  Months. 


Months 

M.   Long. 

Per. 

I 

II 

III 

IV 

V 

N 

VI 

VII 

e  o  •   fr 

„ 

January 

0  0  0  0.0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

February 

1  0  33  18.2 

5 

47 

85 

138 

45 

7 

5 

125 

3 

Maivh    5  ^om- 

1  28  9  11.4 

10 

993 

162 

263 

86 

14 

9 

141 

6 

March    ^  Big 

1  2&  8  19.8 

10 

27 

164 

267 

87 

14 

9 

178 

6 

.  ..     (  Com. 

2  28  42  29.7 

15 

42 

246 

401 

131 

21 

13 

266 

8 

APrU    i  Bis. 

2  29  41  38.0 

15 

76 

249 

405 

132 

21 

13 

302 

8 

myr              (  Com. 

3  28  16  39.6 

20 

59 

329 

534 

175 

28 

18 

355 

11 

May   i  Bis. 

3  29  15  47.9 

20 

92 

331 

538 

176 

28 

18 

391 

11 

Tll_A     j  Com. 

4  28  49  57.9 

26 

110 

414 

672 

220 

35 

22 

480 

14 

June     i  Bis. 

4  29  49  6.2 

26 

144 

416 

676 

221 

35 

23 

516 

14 

T  ,      (  Com. 

5  28  24  7.8 

31 

1-29 

496 

806 

263 

41 

27 

569 

17 

July     \  Bis. 

%  29  23  16.1 

31 

163 

499 

810 

265 

42 

27 

605 

17 

.       (  Com. 

6  28  57  26.1 

36 

182 

580 

943 

309 

49 

31 

694 

20 

AuS'   •  i  Bis. 

6  29  56  34.4 

36 

216 

583 

948 

310 

49 

31 

730 

20 

«  n     (  Com. 

7  29  30  44.2 

41 

233 

665 

81 

354 

56 

36 

819 

23 

SeP'     i  Bis. 

8  0  29  52.6 

41 

268 

668 

86 

355 

56 

36 

855 

23 

o       (  Com. 

8  29  4  54.1 

46 

250 

748 

215 

397 

63 

40 

908 

25 

ifiis. 

9  0  4  2.5 

46 

284 

750 

219 

399 

63 

40 

944 

25 

Nov     5  Com' 

9  29  38  12.5 

51 

300 

832 

353 

443 

70 

45 

33 

28 

Nov'    i  Bis. 

10  0  37  20.7 

51 

333 

835 

357 

444 

70 

45 

69 

28 

D      <  Com. 

10  29  12  22.3 

56 

313 

915 

486 

486 

77 

49 

121 

31 

}Bis. 

11  0  11  30.6 

56 

347 

917 

491 

488 

77 

49 

158 

31 

TABLE  XX. 
Surfs  Motions  for  Days  and  Hours. 


^)ays 

M.  Long. 

Per. 

I 

II 

HI 

IV 

V 

N 

VI 

VII 

Hrs. 

Long. 

I 
VI 

II 

III 

o  /•   " 

// 

/  // 

1 

0  0  0.0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

1 

2  27.8 

1 

0 

0 

2 

0  59  8.3 

0 

34 

3 

4 

1 

0 

0 

36 

0 

2 

4  55.7 

3 

0 

0 

3 

1  58  16.7 

0 

68 

5 

9 

3 

0 

0 

73 

0 

3 

7  23.5 

4 

0 

1 

4 

2  5725.0 

0 

101 

8 

13 

4 

1 

0 

109 

0 

4 

9  51.4 

6 

0 

5 

3  5633.3 

1 

135 

11 

18 

6 

1 

1 

145 

0 

5 

12  19.2 

7 

1 

1 

6 

4  5541.6 

1 

169 

14 

22 

r 

1 

1 

181 

0 

6 

14  47.1 

8 

I 

1 

7 

5  5450.0 

1 

203 

16 

27 

9 

1 

1 

218 

1 

7 

17  14.9 

10 

1 

1 

8 

6  5358.3 

1 

236 

19 

31 

10 

2 

1 

254 

1 

8 

19  42.8 

11 

1 

1 

9 

7  53  6.6 

1 

270 

22 

36 

12 

2 

1 

290 

1 

9 

22  10.6 

13 

1 

2 

10 

8  52  15.0 

1 

304 

25 

40 

13 

2 

1 

327 

1 

10 

24  38.5 

14 

1 

2 

11 

9  51  23.3 

2 

338 

27 

44 

15 

2 

1 

363 

1 

11 

27  6.3 

16 

1 

2 

12 

10  5031.6 

2 

371 

30 

49 

16 

2 

2 

399 

1 

12 

29  34.2 

17 

1 

2 

13 

11  4940.0 

2 

405 

33 

53 

17 

3 

2 

435 

1 

13 

32  2.0 

18 

1 

2 

14 

12  4848.3 

2 

439 

36 

58 

19 

3 

2 

472 

1 

14 

34  29.9 

20 

2 

3 

15 

13  4756.6 

2 

473 

38 

62 

20 

3 

2 

508 

2 

15 

36  57.7 

21 

2 

3 

16 

14  47  4.9 

2 

506 

41 

67 

22 

3 

2 

544 

X* 

16 

39  25.6 

23 

2 

3 

17 

15  46  13.3 

3 

540 

44 

71 

23 

4 

2 

581 

2 

17 

41  53.4 

24 

2 

3 

18 

16  4521.6 

3 

574 

47 

76 

25 

4 

2 

617 

2 

18 

44  212 

25 

2 

3 

19 

17  4429.9 

•3 

608 

49 

80 

26 

4 

3 

653 

2 

19 

46  49.1 

27 

2 

4 

20 

18  43  38.3 

3 

641 

52 

85 

28 

4 

3 

690 

2 

20 

49  16.9 

28 

2 

4 

21 

19  4246.6 

3 

675 

55 

89 

29 

5 

3 

726 

2 

21 

51  44.8 

30 

2 

4 

22 

20  41  54.9 

4 

709 

58 

93 

31 

5 

3 

762 

2 

22 

54  12.6 

31 

2 

4 

23 

21  41  3.3 

4 

743 

60 

98 

32 

5 

3 

798 

2 

23 

56  40.5 

32 

3 

4 

24 

22  40  11.6 

4 

777 

63 

102 

33 

5 

3 

835 

2 

24 

59  8.3 

34 

3 

4 

85 

23  39  19.9 

4 

810 

66 

107 

35 

5 

4 

871 

2 

86 

24  38  28.2 

4 

844 

68 

111 

36 

6 

4 

907 

2 

27 

25  3736.6 

4  878 

71 

116 

38 

6 

4 

943 

2 

38 

26  3644.9 

5 

912 

74 

120 

39 

6 

4 

980 

2 

29 

27  35  53.2 

5 

945 

77 

125 

41 

6 

4 

16 

3 

30 

28  35  1.6 

5 

979 

79 

129  42 

7 

* 

52 

3 

1 

I 

f  ?, 

29  34  9.9 

5   ]  H  ..''•••'•• 

7 

•  :fl   ..-.   .  ' 

i 

TABLE  XXI. 


Sun'*  Motions  for  Minutes  and  Second,. 


TABLE  XXII.  17 

Me™ 


Min. 

Long. 

Min. 

Long. 

Sec. 

Lon 

Sec.  Lon. 

I 

1 

0  2.5 

31 

1  16.4 

1 

0.0 

31 

1.3 

2 

4.9 

32 

1  18.8 

2 

0.1 

32 

1.3 

3 

7.4 

33 

1  21.3 

3 

0.1 

33 

.4 

4 

9.9 

34 

1  23.8 

4 

0.2 

34 

.4 

5 

12.3 

35 

1  26.2 

5 

0.2 

35 

.4 

6 

14.8 

36 

1  28.7 

6 

0.2 

36 

.5 

7 

17.2 

37 

1  31.2 

7 

0.3 

37 

.5 

8 

19.7 

38 

1  33.6 

8 

0.3 

38 

.6 

9 

22.2 

39 

1  36.1 

9 

0.4 

39 

.6 

10 

24.6 

40 

1  38.6 

10 

0.4 

40 

.6 

11 

27.1 

41 

1  41.0 

11 

0.5 

41 

.7 

12 

29.6 

42 

1  43.5 

12 

0.5 

42 

.7 

13 

32.0 

43 

1  46.0 

13 

0.5 

43 

.8 

14 

34.5 

44 

1  48.4 

14 

0.6 

44 

1.8 

15 

37.0 

45 

1  50.9 

15 

0.6 

45 

1.8 

16 

39.4 

46 

1  53.3 

16 

0.7 

46 

1.9 

17 

41.9 

47 

1  55.8 

17 

0.7 

47 

1.9 

18 

44.4 

48 

1  58.3 

18 

0.7 

48 

2.0 

19 

46.8 

49 

2  0.7 

19 

0.8 

49 

2.0 

20 

49.3 

50 

2  3.2 

20 

0.8 

50 

2.0 

21 

51.7 

51 

2  5.7 

21 

0.9 

51 

2.1 

22 

54.2 

52 

2  8.1 

22 

0.9 

52 

2.1 

23 

56.7 

53 

2  10.6 

23 

0.9 

53 

2.2 

24 

59.1 

54 

2  13.1 

24 

1.0 

54 

2.2 

25 

1  1.6 

55 

2  15.5 

25 

1.0 

55 

2.3 

26 

1  4.1 

56 

2  18.0 

26 

1.1 

56 

2.3 

27 

1  6.5 

57 

2  20.5 

27 

1.1 

57 

2.3 

28 

1  9.0 

58 

2  22.9 

28 

1.1 

58 

2.4 

29 

1  11.5 

59 

2  25.4 

29 

1.2 

59 

2.4 

30 

1  13.9 

60 

2  27.8 

30 

1.2 

60 

2.5 

Years  23°27 

1835 

3880 

1836 

38.35 

1837 

37.89 

1838 

37.43 

1839 

36.98 

1840 

36.52 

1841 

36.06 

1842 

35.61 

1843 

35.15 

1844 

34.69 

1845 

34.23 

1846 

33.78 

1847 

33.32 

1848 

32.86 

1849 

32.41 

1850 

31.95 

1851 

31.49 

1852 

31.04 

1853 

30.58 

1854 

30.12 

1855 

29.66 

1856 

29.21 

1857 

2875 

1858 

28'29 

1859 

2784 

1860 

27.38 

1861 

2692 

1862 

26.47 

1863 

26.01 

1864 

25.55 

TABLE  XXIII. 

Sun's  Hourly  Motion. 

Argument.     Sim's  Mean  Anomaly. 


0* 

I* 

II* 

III* 

IV* 

V, 

0 

0 
10 
20 
30 

2  32.92 
2  32.84 
2  32.59 
2  32.20 

2  32.20 
2  31.67 
2  31.02 
2  30.28 

2  30.23 
2  29.46 
2  28.61 
2  27.74 

2  27.74 
2  26.89 
2  26.07 
2  25.32 

2  25.32 
2  24.64 
2  24.06 
2  23.60 

2  23.60 
2  23.26 
2  23.05 
2  22.99 

0 

30 
20 
10 
0 

XI* 

X* 

IX* 

VIII* 

VII* 

VI* 

TABLE  XXIV. 

Sun's  Semi-diameter. 
Argument.    Sim's  Mean  Anomaly. 


0* 

I* 

II* 

III* 

IV* 

V« 

0 

0 
10 
20 
30 

16  17.3 
16  17.0 
16  16.2 
16  15.0 

16  15.0 
16  13.3 
16  11.2 
16  8.8 

16  8.8 
16  6.2 
16  3.4 
16  0.6 

16  0.6 

15  57.8 
15  55.1 
15  52.7 

15  52.7 
15  50.5 
15  48.6 
15  47.0 

15  47.0 
15  45.9 
15  45.2 
15  45.0 

o 
30 
20 
10 
0 

XI, 

X* 

IX* 

VIII* 

VII* 

VI, 

C 

18 


TABLE  XXV. 

Equation  of  the  Surfs  Centre. 
Argument.     Sun's  Mean  Anomaly. 


0 

i« 

Us 

III* 

IV* 

V» 

0 

0 

8       0        '        " 

11  29  59  13.9 

0  5758.5 

1  40  10.7 

1  54  34.1 

1  38    4.8 

0  55  52.6 

1 

0      0    1  1Y.3 

0  59  43.9 

1  41     8.9 

1  54  30.5 

1  37    2.4 

0  54   8.7 

2 

0    3  20.6 

1     1  28.0 

1  42    5.1 

1  54  24.8 

1  3558.1 

0  5224.0 

3 

0    5  23.9 

1     3  10.9 

1  42  59.3 

1  54  17.0 

1  3452.2 

0  50  38.2 

4 

0    7  27.0 

1     4  52.6 

1  43  51.8 

1  54    7.1 

1  3344.6 

04851.6 

5 

0    9  30.0 

1     6  33.0 

1  44  42.1 

1  53  55.2 

1  3235.4 

047   4.2 

6 

0  11  32.8 

1     8  12.3 

1  45  30.4 

1  53  41.0 

1  31  24.4 

0  45  16.0 

7 

0  13  35.4 

1     950.1 

1  46  16.8 

1  53  24.9 

1  30  11.9 

0  43  26.9 

8 

0  15  37.7 

1  11  26.5 

1  47     1.2 

1  53     6.7 

1  28  57.7 

0  41  37.0 

9 

0  17  39.6 

1   13    1.7 

1  47  43.5 

1  52  46.5 

1  2742.0 

03946.5 

10 

0  19  41.2 

1  14  35.3 

1  48  23.9 

1  52  24.2 

1  26  24.8 

0  3755.3 

11 

0  21  42.4 

1  16    7.5 

1  49    2.2 

1  51  59.8 

1  25    5.9 

0  36    3.3 

12 

0  23  43.1 

1  1738.2 

1  49  38.4 

1  51  33.4 

1  2345.7 

0  3410.8 

13 

0  25  43.4 

1   19    7.5 

1  50  12.6 

1  51     5.0 

1  22  23.8 

0  32  17.7 

14 

0  27  43.2 

1  2035.2 

1  30  44.7 

1  50  34.5 

1  21    0.6 

0  3023.8 

15 

0  29  42.3 

1  22    1.5 

1  51  14.9 

1  50  -  2.2 

1   1936.0 

0  2829.6 

16 

0  31  40.9 

1  2326.0 

1  51  42.9 

1  49  27.7 

1   18    9.9 

0  26  34.8 

17 

0  33  38.9 

1  2449.9 

1  52    8.7 

1  48  51.3 

1   1642.4 

0  2439.6 

19 

0  35  36.2 

1  26  10.3 

1  52  32.5 

1  48  13.0 

1   15  13.7 

0  2243.9 

19 

0  37  32.9 

1  2730.0 

1  52  54.3 

1  47  32.7 

1   1343.5 

0  2047.9 

20 

0  39  28.8 

1  2848.0 

1  53  13.9 

1  46  50.4 

1   12  12.1 

0  1851.4 

21 

0  41  23.9 

1  30    4.2 

1  53  31.4 

1  46     6.3 

1   1039.3 

0  1654.6 

22 

0  43  18.1 

1  31  18.8 

1  53  46.8 

1  45  20.3 

1     9    5.4 

0  1457.5 

23 

045  11.5 

1  3231.7 

1  54    0.1 

1  44  32.2 

1     7  30.3 

0  13   0.1 

24 

0  47    4.0 

1  3342.7 

1  54  11.2 

1  43  42.4 

1     5  54.0 

0  11    2.6 

25 

0  48  55.6 

1  3452.0 

1  54  20.4 

1  42  50.7 

1     4  16.5 

0    9    4.8 

26 

0  50  46.3 

1  3559.4 

1  54  27.2 

1  41  57.1 

1     2  37.8 

0    7   6.9 

27 

0  52  36.0 

1  37    5.1 

1  54  32.1 

1  41     1.7 

1     0  58.0 

0    5    87 

28 

0  54  24.6 

1  38    8.8 

1  54  34.9 

1  40    4.5 

0  59  17.3 

0    3105 

29 

0  56  12.1 

1   39  10.8 

1  54  35.4 

1  39     5.6 

0  57  35.4 

0    1  12.2 

L~. 

0  57  58.5 

1  40  10.7 

1  54  34.1 

1  38    4.8 

0  55  52.6 

1 

TABLE  XXVI. 

Secular   Variation  of  Equation  of  Sun's  Centre. 
Argument.    Sun's  Mean  Anomaly. 


0* 

* 

II* 

III* 

IV* 

V* 

0 

0 

—  0 

—  9 

—  15 

—  17 

—  15 

—  8 

2 

1 

9 

15 

17 

14 

8 

4 

1 

10 

16 

17 

14 

7 

6 

2 

10 

16 

17 

14 

7 

8 

2 

11 

16 

17 

13 

6 

10 

3 

11 

16 

17 

13 

6 

12 

4 

12 

17 

17 

12 

5 

14 

4 

12 

17 

16 

12 

5 

16 

5 

13 

17 

16 

12 

4 

18 

5 

13 

17 

16 

11 

3 

20 

6 

13 

17 

16 

11 

3 

22 

7 

14 

17 

16 

10 

2 

24 

7 

14 

17 

15 

10 

2 

26 

8 

15 

17 

15 

9 

1 

28 

8 

15 

17 

15 

9 

1 

30 

—  9 

—  15 

—  17 

—  15 

—  8 

—  0 

TABLE  XXV. 

Equation  of  the  Sun's  Centre- 
Argument.     Sun's  Mean  Anomaly. 


VI* 

VII* 

VIIIS 

IX* 

X» 

s 

11* 

11* 

11* 

11* 

11* 

11, 

0 

0 

29  59  13.9 

29  235.2 

28  20  23.0 

28  353.7 

28  18  17.1 

29  029.3 

1 

29  57  15.6 

29  052.4 

28  19  22.2 

28  3  52.3 

28  19  17.0 

29  2  15.7 

2 

29  55  17.3 

28  59  10.5 

28  18  23.3 

28  3  52.8 

28  20  19.0 

29  4  3.2 

3 

29  53  19.1 

28  57  29.8 

28  1726.1 

28  355.6 

28  21  22.7 

29  5  51.8 

4 

29  51  20.9 

28  55  50.0 

28  16  30.7 

28  4  0.5 

28  22  28.4 

29  741.5 

5 

29  49  23.0 

28  54  11.4 

28  1537.1 

28  4  7.4 

28  23  35.8 

29  932.2 

6 

29  47  25.2 

28  52  33.8 

28  14  45.4 

28  4  16.6 

282445.1 

29  11  23.8 

7 

29  45  27.7 

28  50  57.5 

28  13  55.6 

28  427.7 

28  25  56.1 

29  13  16.3 

8 

29  43  30.3 

28  49  22.4 

28  13  7.5 

28  441.0 

28  27  9.0 

29  15  9.7 

9 

29  41  33.2 

28  47  48.5 

28  12  21.5 

28  456.4 

28  28  23.6 

29  17  3.9 

10 

29  39  36.4 

28  46  15.7 

28  11  37.4 

28  5  13.9 

28  29  39.8 

29  18  59.0 

11 

29  37  39.9 

28  44  44.3 

28  10  55.1 

28  533.5 

28  30  57.8 

29  20  54.9 

12 

29  35  43.9 

28  43  14.1 

28  10  14.8 

28  555.3 

28  32  17.5 

29  22  51.6 

13 

29  33  48.2 

28  41  45.4 

28  9  36.5 

28  6  19.1 

28  33  38.9 

29  24  48.9 

14 

29  31  53.0 

28  40  17.9 

28  9  0.0 

28  644.9 

28  «5  1J8 

29  26  46.9 

15 

29  29  58.2 

28  38  51.8 

28  8  25.6 

28  7  12.9 

28  36  26.3 

29  28  45.5 

16 

2928  4.0 

28  37  27.2 

28  753.2 

28  743.1 

28  37  52.6 

29  30  44.6 

17 

29  26  10.1 

28  36  4.0 

28  7  22.8 

28  8  15.2 

28  39  20.3 

29  32  44-4 

18 

29  24  17.0 

283442.1 

28  654.4 

28  8  49.4 

28  40  49.6 

29  34  44.7 

19 

29  22  24.5 

28  33  21.9 

28  6  28.0 

28  925.6 

28  42  20.3 

29  36  45.4 

20 

29  20  32.5 

28  32  3.0 

28  6  3.6 

28  10  3.9 

28  43  52.5 

29  38  46.6 

21 

29  18  41.3 

28  30  45.8 

28  5  41.4 

28  10  44.3 

284526.1 

29  40  48.2 

22 

29  16  50.8 

28  29  30.1 

28  521.1 

28  11  26.6 

28  47  1.3 

294250.1 

23 

29  15  0.9 

28  28  15.9 

28  5  2.9 

28  12  11.0 

28  48  37.7 

29  44  52.5 

24 

29  13  11.8 

28  27  3.4 

28  446.8 

28  12  57.4 

28  50  15.5 

29  46  55.0 

25 

29  11  23.6 

28  25  52.4 

28  432.6 

28  13  45.7 

28  51  54.8 

29  48  57.8 

26 

29  9  36.2 

28  24  43.2 

28  420.7 

28  14  36.0 

28  53  35.3 

2951  0.8 

27 

29  749.5 

28  23  35.6 

28  4  10.8 

28  15  28.5 

28  55  16.9 

2953  3.9 

28 

29  6  3.8 

28  22  29.7 

28  4  3.0 

28  16  22.7 

28  56  59.8 

29  55  7.2 

29 

29  4  19.1 

28  21  25.4 

28  3  57.3 

28  17  18.9 

28  58  43.9 

29  57  10.5 

30 

29  2  35.2 

28  20  23.0 

28  353.7 

28  18  17.1 

29  029.3 

29  59  13.9 

TABLE  XXVI. 

Secular   Variation  of  Equation  of  Surfs  Centre. 
Argument.    Sun's  Mean  Anomaly. 


VI* 

VII* 

VIII*  • 

IX* 

X* 

XI* 

o 
0 

+  0 

+  8 

+  15 

tr 

+  17 

+  15 

-f  9 

2 

1 

9 

15 

17 

15 

8 

4 

1 

9 

15 

17 

15 

8 

6 

2 

10 

15 

17 

14 

7 

8 

2 

10 

16 

17 

ft 

7 

10 

3 

11 

16 

17 

14 

6 

12 

3 

11 

16 

17 

13 

6 

14 

4 

12 

16 

17 

13 

5 

16 

5 

12 

16 

17 

12 

4 

18 

5 

12 

17 

17 

12 

4 

20 

6 

13 

17 

16 

11 

3 

22 

6 

13 

17 

16 

11 

2 

24 

7 

14 

17 

16 

10 

*> 

m 

26 

7 

14 

17 

16 

10 

i 

28 

8 

14 

17 

15 

9 

i 

30 

+  8 

+  15 

+  17 

+  15 

+  9 

+  0 

TABLE  XXVII. 


Nutations. 
Argument.     Supplement  of  the  Node,  or  N. 


Solar  Nutation, 


N. 

Long. 

R.  Asc. 

Obliq. 

N. 

Long. 

R.  Asc. 

Obliq. 

Long.  Obliq. 

0 

+  0.0 

+0.0 

+  9.2 

500 

—  0.0 

—  0.0 

—  9.3 

Jan. 

ff 

fr 

10 

1.0 

1.0 

9.1 

510    1.1 

1.0 

9.3 

1 

+  0.5  —0.5 

20 

2.1 

2.1 

9.1 

520    2.2 

2.0 

9.3 

11 

0.8 

0.4 

30 

3.2 

3.0 

9.0 

530 

3.3 

2.9 

9.2 

21 

1.1 

0.2 

40 

4.2 

4.0 

8.9 

540 

4.4 

3.9 

9.0 

31 

1.2 

—  0.1 

50 

+  5.2 

+  4.9 

+  8.7 

550 

—  5.5 

—  4.8 

—  8.9 

Feb. 

60 

6.2 

6.0 

8.5 

560 

6.5 

5.7 

8.7 

10 

1.2 

+  0.1 

70 

7.2 

6.9 

8.3 

570 

7.5 

6.6 

8.4 

20 

1.0 

0.3 

80 

8.2 

7.8 

8.1 

580 

8.5 

7.5 

8.1 

90 

9.1 

8.7 

7.8 

590 

9.5 

8.4 

7.8 

March. 

100 

+  10.0 

+  9.4 

4-  7.5 

600 

—  10.4 

—  9.1 

—  7.5 

2 
12 

0.7 
+  0.3 

0.5 

110 

10.8 

10.3 

7.1 

610 

11.2 

9.9 

7.1 

22 

—  0.1 

0.5 

120 

11.6 

11.  1 

6.7 

620 

12.0 

10.6 

6.7 

130 

12.4 

11.7 

6.3 

630 

12.8 

11.4 

6.3 

April. 
i 

OC 

A  C 

140 
150 

13.1 
+  13.8 

12.4 
+  13.01 

5.9 

+  5.5 

640 
650 

13.5 

—  14.2 

12.0 
—  12.6 

5.9 
—  5.4 

i 
11 
21 

.*J 

0.8 
1.1 

u.o 

0.2 
0.2 

160 

14.4 

13.6 

5.0 

660 

14.8 

13.2 

4.9 

TVT 

170 

15.0 

14.1 

4.5 

670 

15.3 

13.8 

44 

May. 

180 
190 

15.5 
15.9 

14.5 

14.8 

4.0 
3.5 

680 
690 

15.8 
16.2 

14.2 
14.7 

3.9 
3.3 

1 
11 

91 

1.2 
1.2 

I  1 

+  0.1 
—  0.1 
0  ^ 

200 

+  16.3 

+  15.1 

+  2.9 

700 

—  16.6 

—  15.0 

—  2.8 

£  1 

31 

1  .  1 

0.8 

U.O 

0.4 

210 

16.6 

15.4 

2.4 

710 

16.9 

15.3 

2.2 

220 
230 
240 
250 

16.9 
17.1 
17.2 
+  17.3 

15.6 
15.7 
15.9 
+  15.9 

1.8 
1.2 
0.7 

+  0.1 

720 
730 
740 
750 

17.1 

17.2 
17.3 
—  17.3 

15.4 
15.7 
15.9 
—  15.9 

1.6 
1.1 
—  0.5 

+  0.1 

June. 
10 
20 
30 

0.4 
—  0.0 
+  0.4 

0.5 
0.5 
0.5 

260 
270 

17.3 
17.2 

15.9 
15.7 

—  0.5 

l.l 

760 
770 

17.2 
17.1 

15.9 
15.7 

0.7 

1.2 

July. 
10 

0.7 
1  0 

0.4 

0  *-l 

280 

17.1 

15.6 

1.6 

780 

16.9 

15.4 

J.8 

J™ 

U.O 

01 

290 

16.9 

15.4 

2.2 

790 

16.6 

15.3 

2.4 

30 

1.2 

.1 

300 

+  16.6 

+  15.1 

_  2.8 

800 

_16.3 

-15.0 

+  2.9 

Aug. 

310 

16.2 

14.8 

3.3 

810 

15.9 

14.7 

3.^ 

9 

1.3 

1  9 

+  0.0 

ft  4. 

320 
330 

15.8 
15.3 

14.5 
14.1 

3.9 
4.4 

820 
830 

15.5 
15.0 

14.2 

13.8 

4.0 

4.5 

29 

i  .  At 

0.9 

U.fc 

0.4 

340 

14.8 

13.6 

4.9 

840 

14.4 

13.2 

5.0 

Sept. 

350 

+  14.2 

4.  13.0 

—  5.4 

850 

—  13.8 

—  12.6 

+  5.5 

8 

0.6 

0.5 

360 
370 

13.5 

12.8 

12.4 
11.7 

5.9 
6.3 

860 
870 

13.1 
12.4 

12.0 
11.4 

5.9 
6.3 

18 
28 

+  0.2 
—  0.2 

0.5 
0.5 

380 

12.0 

11.1 

6.7 

880 

11.6 

10.6 

6.7 

Oct. 

390 

11.2 

10.3 

7.1 

890 

10.8 

9.9 

7.1 

8 

0.6 

0.5 

400 

-}-  10-4 

+  9.4 

_  7.5 

900 

—  10.0 

—  9.1 

+  7.5 

18 

1.0 

0.3 

OQ 

1.2 

0.2 

410 

9.5 

8.7 

7.8 

910 

9.1 

8.4 

7.8 

ivo 

420 

8.5 

7.8 

8.1 

920 

8.2 

7.5 

8.1 

Nov. 

430 

7.5 

6.9 

8.4 

930 

7.2 

6.6 

8.3 

7 

1.2 

+  0.0 

440 

6.5 

6.0 

8.7 

940 

6.2 

5.7 

8.5 

17 

1.2 

0.2 

450 

+  5.5 

+  4.9 

—  8.9 

950 

—  5.2 

—  4.8 

+  8.7 

27 

1.0 

0.4 

460 

4.4 

4.0 

9.0 

960 

4.2 

3.9 

8.9 

Dec. 

470 

3.3 

3.0 

9.2 

970 

3.2 

2.9 

9.0 

7 

0.6 

0.5 

480 

2.2 

2.1 

9.3 

980 

2.1 

2.0 

9.1 

17 

—  0.2 

0.5 

490 

i.r 

1.0 

9.3 

990 

1.0 

1.0    9.1 

27 

+  0.3 

0.5 

500 

+  0.0 

+  0-0 

—  9.3 

1000 

—  0.0 

—  0.01+  2.2 

37 

+  0.6 

—  0.5,' 

TABLE   XXVIII. 


TABLE  XXIX, 


2J 


Lunar  Equation,  1st  part. 
Argument  I. 


Lunar  Equation,  2d  part. 
Arguments  I.  and  VI. 
I. 


I 

EqU 

I 

Equ 

0 

7. 

500 

7.5 

10 

8. 

510 

7.0 

20 

8. 

520 

6.6 

30 

8. 

530 

6.1 

40 

9.4 

540 

5.6 

50 

9.8 

550 

5.2 

60 

10.3 

560 

4.7 

70 

10.7 

570 

4.3 

80 

11. 

580 

3.9 

90 

11.5 

590 

3.5 

100 

11.9 

600 

3.1 

110 

12.3 

610 

2.7 

120 

12.6 

620 

2.4 

130 

13.0 

630 

2.0 

140 

13.3 

640 

1.7 

150 

13.6 

650 

1.4 

160 

13.8 

660 

1.2 

170 

14.1 

670 

0.9 

180 

14.3 

680 

0.7 

190 

14.5 

690 

0.5 

200 

14.6 

700 

0.4 

210 

14.8 

710 

0.2 

220 

14.9 

720 

0.1 

230 

14.9 

730 

0.1 

240 

15.0 

740 

0.0 

250 

15.0 

750 

0.0 

260 

15.0 

760 

0.0 

270 

14.9 

770 

0.1 

280 

14.9 

780 

0.1 

290 

14.8 

790 

0.2 

300 

14.6 

800 

0.4 

310 

14.5 

810 

0.5 

320 

14.2 

820 

0.7 

330 

14.1 

830 

0.9 

340 

13.8 

840 

1.2 

350 

13.6 

850 

1.4 

360 

13.3 

860 

1.7 

370 

13.0 

870 

2.0 

380 

12.6 

880 

2.4 

390 

12.3 

890 

2.7 

400 

11.9 

900 

3.1 

410 

11.5 

910 

3.5 

420 

11.1 

920 

3.9 

430 

10.7 

930 

4.3 

440 

10.3 

940 

4.7 

450 

9.8 

950 

5.2 

460 

9.4 

960 

5.6 

470 

8.9 

970 

6.1 

480 

84 

980 

6.6 

490 

8.0 

990 

7.0 

600  7.5 

000 

7.5 

VI 

0 

50 

100 

150  200 

25C 

30C 

350 

fiOO 

450 

500 

0 

.3 

1.2 

1.2 

1.1  !  1-0 

1.0 

1.0 

1.1 

1.2 

1.2 

1.3 

50 

.5 

1.5 

1.5 

1.3  1.1 

1.0 

0.9 

1.0 

1.1 

1.1 

1.1 

100 

.7 

1.8 

1.7 

1.4  1.2 

1.1 

1.0 

0.9 

0.9 

0.9 

0.9 

150 

.9 

1.9 

1.8 

1.6  1.4 

1.3 

.0 

0.8 

0.8 

0.8 

0.7 

200 

.9 

2.0 

2.0 

1.711.5 

1.4 

.0 

0.8 

0.8 

0.8 

0.7 

250 

2.0 

2.0 

2.0 

1.8  i  1.6 

1.5 

.1 

0.9 

0.7 

0.7 

0.6 

300 

1.9 

1.9 

1.9 

1.9  !  1.7 

1.6 

.2 

1.0 

0.8 

0.7 

0.7 

350 

1.8 

1.9 

1.9 

1.9  1.7 

1.6 

.4 

1.0 

1.0 

0.9 

0.8 

400 

1.6 

1.7 

1.8 

1.9  '  1.7 

1.6 

.4 

1.2 

1.1 

1.0 

1.0 

450 

1.5 

1.5 

1.6 

1.7  1.7 

1.7 

.6 

1.4 

.2 

1.2 

1.1 

500 

1.3 

1.4 

1.4 

1.5!  1.7 

1.7 

.7 

1.5 

.4 

1.4 

1.3 

i 

550 

1.1 

1.2 

1.2 

1.4  '  1.6 

1.7 

ft 

}  7 

.6 

1.5 

1.5 

600 

1.0 

1.0 

1  lil.tjl.4 

1.6 

.8 

1.8 

.8 

1.7 

1.6 

650 

0.8 

0.9 

1.0 

1.1 

1.3 

1.5 

.7 

1.8 

.9 

1.9 

1.8 

700 

0.7 

0.7 

0.8 

1.1 

1.2 

1.4 

.7 

1.9 

.9 

1.9 

1.9 

750 

0.6 

0.6 

0.7 

1.0 

1.1 

T.3 

.6 

1.9 

.9 

2.0 

2.0 

800 

0.7 

0.7 

0.7 

0.9 

1.1 

1.2 

.5 

1.8 

2.0 

.9 

1.9 

850 

0.7 

0.8 

0.8 

0.9 

0.9 

1.1 

.4 

1.7 

.8 

.8 

1.9 

900 

0.9 

0.9 

0.9 

0.9 

1.0 

1.1 

.2 

1.5 

.7 

.7 

1.7 

950 

1.1 

1.0 

1.1 

1.0 

1.0 

1.0 

.1 

1.3 

.4 

.6 

1.5 

0 

1.3 

1.2 

1.2 

1.1 

1.0 

1.0 

.0 

1.1 

1.2 

.2 

1.3 

I. 

VI 

500 

550 

600 

650 

700 

750 

800 

850 

900 

950 

1000 

0 

1.3 

1.4 

1.4 

1.5 

1.6 

1.6 

1.6 

1.5 

1.4 

.4 

1.3 

50 

1.1 

1.1 

1.2 

1.3 

1.5 

1.5 

1.7 

1.6 

1.5 

.5  1.5 

100 

0.9 

0.9 

0.9 

1.1 

1.3 

1.5 

1.6 

1.7 

1.7 

.7  1.7 

150 

0.7 

0.8 

0.810.9 

1.2 

1.4 

1.6 

1.9 

1.8 

.8  1.9 

200 

0.7 

0.7 

0.6  0.8 

1.1 

1.2 

1.6 

1.8 

.8 

.8 

1.9 

250 

0.6 

0.6 

0.7 

0.7 

1.0 

1.1 

1.5 

1.7 

.9 

.9 

2.0 

300 

0.7 

0.7 

0.7 

0.7 

0.9 

1.0 

1.4 

1.6 

.8 

.9 

1.9 

350 

0.8 

0.7 

0.7 

0.8 

0.9 

1.0 

1.4 

1.6 

.6 

.7 

1.8 

400 

1.0 

0.9 

0.8 

0.8 

0.9 

1.0 

1.2 

1.4 

.5 

.6 

1.6 

450 

1.1 

1.1 

1.0 

0.9 

0.9 

0.9 

1.0 

1.2 

.4 

.4 

1.5 

500 

1.3 

1.2 

1.2 

1.1 

0.9 

0.9 

0.9 

1.1 

1.2 

1.2 

1.3 

550 

1.5 

1.4 

1.4 

1.2 

1.0 

0.9 

0.9 

0.9 

1.0 

1.1 

i.i 

600 

1.6 

1.6 

1.5 

1.4 

1.2 

1.0 

0.8 

0.8 

0.8 

0.9 

1.0 

650 

1.8 

1.7 

1.6 

1.6 

1.3 

.1 

0.9 

0.8 

0.7  0.7 

0.8 

700 

1.9 

1.8 

1.8 

1.6 

1.4 

.2 

0.9 

0.7 

0.7 

0.7 

0.7 

750 

2.0 

1.9 

1.9 

1.7 

1.5 

.3 

1.0 

0.7 

0.7 

0.6 

0.6 

800 

1.9 

1.8 

1.8 

1.8 

1.6 

.4 

1.1 

0.8 

0.6 

0.7 

0.7 

850 

1.9 

1.8 

1.8 

1.8 

1.6 

.5 

1.2 

0.9 

0.8 

0.8 

0.7 

900 

1.7 

1.7 

1.7 

1.7 

1.6 

1.5 

1.3 

1.1 

0.9 

0.9 

0.9 

950 

1.5 

1.5 

1.5 

1.6 

1.7 

1.6 

1.5 

1.3 

1.2 

1.1 

1.1 

0 

1.3 

1.4 

1.4 

1.5 

1.6 

1.6 

1.6 

1.5 

1.4 

1.4 

1.3 

Constant  1".3. 

TABLE  XXX. 


Perturbations  produced  by  Venus. 

Arguments  II  and  III. 

III. 


fll. 

0 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120' 

—  1 

" 

" 

0 

21.6 

20.8 

19.8 

19.0 

17.9 

16.8 

15.9 

14.7 

14.0 

13.2 

128  125 

12.9 

20  J23.1 

22.7 

21.6 

21.0 

20.1 

19.3 

18.4 

17.4 

16.4 

15.5 

14.5  13.8 

13.4 

40  23.5 

23.2 

22.9 

22.7 

22.0 

21.1 

20.4 

19.5 

18.7 

17.9 

16.9  16.1 

15.3 

60  !  22.2 

22.5 

23.1 

22.7 

22.8 

22.5 

21.9 

21.3 

20.5 

19.9 

19.1 

18.2 

17.4 

80  !  20.0 

20.7 

21.4 

21.7 

22.1 

22.3 

22.2 

22.2. 

21.7 

21.3 

20.7 

19.9 

19.3 

100 

17.6 

18.6 

19.2 

19.9 

20.5 

21.0 

21.6 

21.7 

21.6 

21.6 

21.5 

21.1 

20.5 

120 

15.3 

16.0 

16.9 

17.7 

18.4 

19.2 

19.8 

20.2 

20.7 

20.8 

21.1 

21.1 

20.8 

140 

13.6 

14.2 

14.8 

15.5 

16.2 

17.0 

17.6 

18.3 

19.0 

19.4 

20.0 

20.0 

20.4 

160 

12.7 

13.2 

13.6 

14.1 

14.6 

15.0 

15.7 

16.4 

17.0 

17.3 

18.1 

18.7 

19.2 

180 

12.7 

12.9 

13.1 

13.5 

13.9 

14.0 

14.5 

14.8 

15.0 

15.8 

16.4 

16.8 

17.2 

200 

13.2 

13.2 

13.2 

13.4 

13.7 

13.8 

14.1 

14.2 

14.5 

14.5 

14.8 

15.2 

16.0 

220 

13.5 

13.6 

13.9 

14,1 

14.1 

14.1 

14.2 

14.3 

14.5 

14.6 

14.6 

14.7 

14.8 

240 

13.6 

13.8 

14.1 

14.4 

14.6 

14.8 

14.8 

14.9 

15.1 

15.1 

15.1 

14.9 

14.8 

260 

12.8 

13.3 

13.8 

14.2 

14.6 

15.0 

15.3 

15.6 

15.5 

15.5 

15.6 

15.6 

15.6 

280 

11.5 

12.3 

13.0 

13.4 

14.0 

14.6 

15.1 

15.4 

16.0 

16.2 

16.2 

16.3 

16.2 

300 

10.1 

10.9 

11.3 

12.1 

12.9 

13.7 

14.2 

14.9 

15.4 

16.0 

16.4 

16.5 

16.7 

320 

8.2 

8.8 

9.6 

10.6 

11.3 

12.0 

12.9 

13.7 

14.3 

15.0 

15.8 

16.3 

16.8 

340 

6.9 

7.5 

8.1 

8.4 

9.4 

10.1 

11.1 

11.9 

12.7 

13.6 

14.4 

15.2 

16.0 

360 

6.5 

6.5 

6.8 

7.4 

8.0 

8.4 

9.1 

9.9 

10.8 

11.5 

12.6 

13.4 

14.4 

380 

6.8 

6.5 

6.3 

6.4 

6.7 

7.0 

7.6 

8.2 

8.9 

9.6 

10.6 

11.4 

12.4 

400 

7.5 

7.1 

6.7 

6.4 

6.2 

6.4 

6.5 

6.9 

7.5 

7.9 

8.7 

9.4 

10.3 

420 

9.1 

8.4 

7.6 

7.1 

6.7 

6.5 

6.3 

'6.2 

6.7 

6.8 

7.2 

7.8 

8.4 

440 

10.6 

9.8 

9.0 

8.6 

7.9 

7.2 

6.7 

6.4 

6.4 

6.4 

6.6 

6.8 

7.1 

460 

12.1 

11.5 

10.5 

9.6 

9.0 

8.5 

8.0 

7.3 

6.8 

6.6 

6.5 

6.4 

6.5 

480 

13.6 

12.8 

11.9 

11.0 

10.4 

9.6 

8.8 

8.2 

7.7 

7.2 

6.8 

6.4 

6.5 

500  15.1 

14.4 

13.4 

12.4 

11.6 

10.8  10.1 

9.3 

8.6 

8.1 

7.5 

7.1 

6.8 

520  16.5  15.6  14.8 

13.9 

13.1 

12.3  'll.3 

10.5 

9.7 

9.1 

8.6 

7.9 

7.4 

540 

18.1  17.5 

16.4 

15.5 

14.5 

13.7;i2.8 

11.8 

11.1 

10.4 

9.7 

8.9 

8.2 

560 

20.4  19.3 

18.2 

17.6 

16.5 

15.4  14.4 

13.4 

12.7 

11.6 

10.8 

10.2 

9.2 

580 

22.8  j  21.7  20.7 

19.7 

18.4 

17.6  16.6 

15.5 

14.3 

13.4 

12.5 

11.6 

10.6 

600 

25.2  24.1 

23.1 

22.2 

21.2 

19.9  :  18.6 

17.8 

16.6 

15.6 

14.5 

13.4 

12.6 

620 

27.3  26.5 

25.6 

24.7 

23.5 

22.5)21.6 

20.4 

19.0 

18.1 

16.8 

15.7 

14.7 

640 

29.0  28.5 

27.7 

26.9 

26.2 

25.1 

24.1 

22.9 

21.8 

20.8 

19.6 

18.4 

17.2 

660 

29.8  29.6 

29.2 

28.5 

28.1 

27.4 

26.5 

25.6 

24.5 

23.4 

22.5 

21.2 

19.8 

680 

29.7  29.6 

29.5 

29.5 

29.1 

28.8  28.2 

27.6 

27.0 

26.0 

25.0 

23.8 

22.8 

700 

28.8  29.2 

29.3 

29.5 

29.5 

29.5  29.2 

28.8 

28.4 

27.8 

27.2 

26.4 

25.2 

720 

26.9  27.6 

28.3 

29.0 

29.2 

29.4  29.4 

29.3 

29.1 

28.9 

28.4 

27.9 

27.3 

740 

24.7  25.7 

26.6 

27.3 

27.9 

28.5  29.1 

29.0 

29.2 

29.3 

29.1 

28.8 

28.4 

760 

22.2  23.5 

24.3 

25.3 

26.2 

27.0  j  27.6 

28.3 

28.6 

28.7 

28.9 

29.1 

29.0 

780 

19.6  21.0 

22.0 

23.2 

24.2 

25.1  25.9 

26.7 

27.3 

27.8 

28.4 

28.5 

28.7 

800 

17.2  18.5 

19.3 

20.9 

21.8 

22.9 

23.9 

25.0 

25.8 

26.4 

26.9 

27.6 

28.1 

820 

15.2  15.9 

17.0 

18.4 

18.9 

20.7 

21.7 

22.8 

23.8 

24.8 

25.6 

26.2 

26.6 

840 

13.2  14.0 

15.0 

16.0 

17.0 

18.2 

18.8 

20.3 

21.7 

22.7 

23.6 

24.5 

25.3 

860 

11.5  12.2 

13.0 

13.9 

14.9 

15.9  17.1 

18.0 

18.9 

20.3 

21.4 

22.6 

23.5 

880 

11.0  11.2 

11.5 

12.2 

13.0 

13.7  14.8 

15.7 

16.8 

18.1 

19.1 

20.2 

21.1 

900 

11.2  10.2 

10.9 

11.5 

12.5 

12.1 

12.8 

13.7 

14.5 

15.5 

16.6 

17.9 

18.5 

920 

12.1  11.6 

11.5 

11.1 

11.2 

11.3 

11.7 

12.1 

12.7 

13.4 

14.4 

15.2 

16.4 

940 

14.0  13.3 

12.6:  12.3 

11.6 

11.5 

11.3 

11.4 

11.6 

12.0 

12.8 

13.3 

14.2 

960 

16.7  15.6 

14.6  13.7 

13.1 

12.5 

11.9 

11.7 

11.6 

11.4 

11.7 

12.1 

12.6 

980 

19.5  18.3 

17.3 

16.4 

15.2 

14.2 

13.4 

12.7 

12.2 

12.0 

11.9 

11.8 

11.8 

1000 

21,6  20.8 

19.8 

19.0 

17.9 

16.8 

15.9 

14.7 

14.0 

13.2. 

12.8 

12.5 

12.2 

0 

10 

20 

30 

40 

59 

60 

70 

80 

90 

00 

110 

120 

TABLE  XXX. 


Perturbations  produced  by  Venus, 

Arguments  II  and  III. 

HI. 


II. 

120  130 

140 

150  i  160 

170 

180  |  190  200 

210 

220 

230 

240 

„    „    „ 

0 

12.2  |  12.2 

12.3 

12.4  12.8 

13.3 

13.9 

14.7  15.6 

16.5 

17.7 

18.8 

20.1 

20 

13.4  12.9 

12.6 

12.3  :  12.2 

12.4 

12.9 

13.3;  14.0 

14.6 

15.5 

16.4 

17.3 

40 

15.3  14.4 

14.0 

13.5:13.0  12.9 

12.6 

12.6  13.1 

13.5 

14.0 

14.4 

15.4 

60 

17.4  16.7 

16.0 

15.2  14.5 

14.0 

13.6 

13.3  13.2 

13.2 

13.4 

13.5 

14.1 

80 

19.3118.7 

17.7 

17.1  16.4 

15.9 

15.4 

14.6  14.3 

13.9 

13.8 

13.7 

13.6 

100 

20.5  j  20.2 

19.5 

18.9 

18.2 

17.5 

17.1 

16.3 

15.9 

15.4 

14.8 

14.6 

14.3 

120 

20.8  I  20.7 

20.4 

20.0 

19.7 

19.2 

18.5 

18.0 

17.3 

16.9 

16.5 

16.2 

15.6 

140 

20.4 

20.4 

20.2 

20.0 

20.1 

19.7 

19.5 

19.3 

18.8 

18.2 

17.7 

17.4 

17.0 

160 

19.2 

19.1 

19.4 

19.7 

19.5 

19.6 

19.3 

19.6 

19.2 

19.0 

18.7 

18.4 

18.1 

180 

17.2 

17.7 

18.5 

18.5 

18.5 

18.8 

18.4 

18.8 

19.0 

19.0 

18.9 

18^6 

18.5 

200 

16.0 

16.2 

16.6 

16.8 

17.5 

17.6 

17.7 

17.9 

18.1 

18.2 

18.3 

18.3 

18.3 

220 

14.8 

15.0 

15.3 

15.7 

16.1 

16.2 

16.6 

16.8 

17.1 

17.5 

17.1 

17.4 

17.5 

240 

14.8 

14.7 

14.8 

15.0 

15.1 

15.4 

15.7 

15.8 

16.0 

16.1 

16.1 

16.3 

16.4 

260 

15.6 

15.7 

15.3 

14.8 

15.0 

15.0 

15.1 

15.0 

15.1 

15.2 

15.2 

15.1 

15.3 

280 

16.2 

16.2 

16.2  15.9 

15.8 

15.8 

15.5 

15.4 

15.1 

14.9 

14.8 

14.7 

15.0 

300 

16.7 

17.0 

17.1 

16.9 

16.9 

1G.6 

16.5 

16.3 

15.9 

15.7 

15.2 

14.9 

14.8 

320 

16.8 

17.3 

17.5 

17.6 

17.7 

.17.6 

17.5 

17.2 

17.0 

16.8 

16.5 

16.1 

15.6 

340 

16.0 

16.4 

17.2 

IV.  8 

17.9 

18.1 

18.3 

18.2 

18.2 

17.9 

17.5 

17.3 

16.8 

360 

14.4 

15.2 

16.0 

16.7 

17.4 

18.1 

18.4 

18.6 

18.8 

18.8 

18.8 

18.7 

18.4 

380 

12.4 

13.4 

14.3 

15.3 

16.1 

16.9 

17.5 

18.1 

18.6 

19.1 

19.3 

19.5 

19.5 

400 

10.3 

11.2 

12.3 

13.2 

14.2 

15.1 

16.0 

16.8 

17.8 

18.4 

18.8 

19.3 

19.8 

420 

8.4 

9.2 

10.0 

11.0 

12.2 

13.0 

14.1 

15.0 

15.9 

16.9 

17.7 

18.5 

19.0 

440 

7.1 

7.6 

8.4 

9.0 

9.9 

10.9 

11.8 

12.9 

13.8 

14.9 

16.0 

16.7 

17.8 

460 

6.5 

6.8 

7.2 

7.4 

8.1 

9.0 

9.7 

10.6 

11.7 

12.6 

13.8 

14.6 

15.9 

480 

6.5 

6.5 

6.4 

6.6 

7.0 

7.5 

8.2 

8.8 

9.6 

10.4 

11.5 

12.5 

13.5 

500 

6.8 

6.7 

6.5 

6.3 

6.5 

6.6 

7.0 

7.4 

8.2 

8.6 

9.4 

10.4 

11.3 

520 

7.4 

7.0 

6.8 

6.5 

6.3 

6.1 

6.3 

€.6 

7.0 

7.5 

8.0 

8.8 

9.3 

540 

8.2 

7.6 

7.2 

6.8 

6.5 

6.3 

6.2 

6.0 

6.2 

6.5 

6.9 

7.4 

7.9 

560 

9.2 

8.6 

7.9 

7.5 

6.8 

6.6 

6.3 

6.1 

6.0 

6.1 

6.2 

6.5 

6.9 

580 

10.6 

9.8 

9.1 

8.4 

7.7 

7.3 

6.6 

6.3 

6.1 

5.9 

5.7 

5.9 

6.0 

,600 

12.6 

11.4 

10.5 

9.5 

8.7 

8.1 

7.4 

7.0 

6.4 

6.1 

5.8 

5.5 

5.6 

620 

14.7 

13.5 

12.4 

11.4 

10.4 

9.5 

8.7 

7.9 

7.3 

6.7 

6.2 

5.6 

5.2 

640 

17..2 

16.2 

14.9 

13.7 

12.5 

11.4 

10.4 

9.5 

8.7 

7.8 

7.0 

6.5 

5.9 

660 

19.8 

19.0 

17.6 

16.5 

15.1 

13.9 

12.8 

11.5 

10.5 

9.6 

8.6 

7.7 

6.9 

680 

22.8 

21.7 

20.4 

19.3 

18.1 

16.8 

15.7 

14.2 

13.0 

11.9 

10.7 

9.6 

8.6 

700 

25.2 

24.3 

23.3 

22.1 

20.7 

19.7 

18.5 

17.3 

16.0 

14.3 

13.4 

12.1 

11.0 

720 

27.3 

26.4 

25.7 

24.5 

23.7 

22.5 

21.1 

20.2 

18.8 

17.7 

16.4 

15.3 

13.9 

740 

28.4 

27.7 

27.4 

26.6 

25.9 

24.9 

24.0- 

22.8 

21.5 

20.6 

19.2 

18.1 

16.8 

760 

29.0 

28.7 

28.3 

27.8 

27.3 

26.8 

25.9 

25.2 

24.3 

23.0 

21.7 

20.7 

19.7 

780 

28.7 

28.7 

28.8 

28.7 

28.3 

28.0 

27.2 

26.1 

26.1 

25.2 

24.3 

23.3 

22.2 

800 

28.1 

28.3 

28.4 

28.5 

28.5 

28.4 

28.2 

27.3 

27.3 

26.7 

25.9 

25.1 

24.4 

820 

26.6 

27.3 

27.8 

26.1 

28.3 

28.1 

28.1 

28.0 

27.9 

27.7 

27.2 

26.5 

25.9 

840 

25.3 

26.2 

26.7 

27.2 

27.5 

27.9 

28.1 

28.1 

27.9 

27.9 

27.6 

27.3 

27.2 

860 

23.5 

24.5 

25.1 

25.9 

26.6 

27.1 

27.4 

27.7 

27.9 

28.0 

27.9 

27.7 

27.5 

880 

21.1 

22.4 

23.3 

24.2 

25.1 

25.8 

26.5 

27.0 

27.3 

27.5 

27.8 

28.0 

27.7 

900 

18.5 

20.1 

21.3 

22.1 

23.1 

24.7 

25.0 

25.7 

26.3 

26.9 

27.3 

27.5 

27.6 

920 

16.4 

17.7 

18.4 

20.0 

21.0 

22.2 

23.0 

23.9 

24.9 

25.7 

26.2 

26.9 

27.3 

940 

14.2 

14.9 

16.1 

17.5 

18.2 

19.6 

20.8 

21.9 

23.0 

23.9 

24.7 

25.7 

26.1 

960 

12.6 

13.3 

14.1 

14.4 

15.9 

17.2 

17.9 

19.5 

20.5 

21.7 

22.7 

23.9 

24.7 

980 

11.8 

12.1 

12.7 

13.3 

14.1 

14.8 

15.6 

16.8 

17.6 

19.3 

20.2 

21.4 

22.6 

1000 

12.2 

12.2 

12.3 

12.4 

12.8 

13.3 

13.9 

14.7 

15.6 

16.5 

17.6 

8.8 

20.1 

120 

130 

140 

150 

160 

170 

180 

190 

200 

210 

220 

230 

24) 

TABLE  XXX. 


Perturbations  produced  by   Venus. 

Arguments  II.  and  III. 

III. 


II. 

}  240 

250 

260 

270  280 

290 

300 

310 

320 

330 

340 

350  360 

0 

20.1 

21.1 

22.2 

23.4  24.3 

25.2 

25.8 

26.6 

27.2 

27.6 

27.7 

27.G 

27.6 

20 

17.3 

18.6 

19.7 

20.9  21.9 

23.0 

24.2 

24.9 

25.8 

26.6 

27.0 

27.4  !  27.7 

40 

15.4 

16.5 

17.3 

18.3  19.4 

20.5 

21.6 

22.7 

23.7 

24.9 

25.5 

26.3  26.9 

60 

14.1 

14.6 

15.2 

16.3,  17.2 

18.1 

18.9 

20.3 

21.2 

22.3 

23.4 

24.5  !  25.3 

80 

13.6 

14.0 

14.5 

14.9  15.5 

lu.3 

17.3 

18.2 

19.0 

20.0 

21.1 

22.0123.1 

100 

14.3 

14.3 

14.3 

14.4  14.6 

15.0 

15.5 

16.2 

16.9 

17.7 

18.9 

19.8  20.8 

120 

15.6 

15.2 

14.8 

14.8  15.0 

14.9 

15.0 

15.2 

15.9 

16.3 

17.0 

17.7  18.5 

HO  1  17.0 

1G.6  16.4 

15.8  15.5 

15.4 

15.6 

15.6 

15.5 

15.6 

16.1 

16.7  17.1 

160  !  18.1 

17.7  17.5 

17.3  16.9 

16.6 

16.3 

15.9 

16.1 

16.3 

16.3 

16.2!  16.5 

180 

18.5 

18.5  18.3 

18.1 

17.9 

17.6 

17.5 

17.3 

17.0 

16.9 

16.7 

16.8!  16.9 

200 

18.3 

18.4 

18.2 

18.2 

18.2 

18.2 

18.1 

18.1 

17.8 

17.7 

17.6 

17.5 

17.7 

220 

17.5 

17.6 

17.8 

17.8 

18.0 

18.0 

18.2 

18.1 

18.1 

18.3 

18.4 

18.3 

18.3 

240 

16.4 

16.5 

16.7 

16.9 

17.1 

17.3 

17.3 

17.7 

17.5 

18.0 

18.3 

18.4  18.6 

260  15.3 

15.5 

15.5 

15.6 

15.8 

16.1 

16.4 

16.6 

16.8 

16.9 

17.4 

17.7 

18.2 

280  l  15.0 

14.9 

14.9 

14.9 

14.9 

14.7 

15.0 

15.3 

15.5 

15.9 

16.1 

16.4 

16.8 

300  14.8 

14.6  14.6 

14.2 

14.0 

14.0 

13.9 

13.9 

14.2 

14.5 

14.8 

15.0 

15.5 

320  15.6 

15.3 

14.7 

14.5 

14.4 

13.1 

13.6 

13.4 

13.3 

13.1 

13.4 

13.6 

13.8 

340 

16.8 

16.6 

16.0 

15.5 

15.2!  14.5 

14.3 

13.7 

13.1 

13.0 

12.7 

12.6 

12.6 

360 

18.4 

17.9 

17.5 

17.0 

16.5 

15.9 

15.4 

14.9 

14.3 

13.7 

13.0 

12.6 

12.3 

380 

19.5 

19.2 

18.9 

18.5 

17.9 

17.7 

16.9 

16.4 

15.8 

15.0 

14.5 

13.6 

13.1 

400 

19.8 

19.8 

20.1 

19.7 

19.4 

19.1 

18.6 

18.1 

17.5 

17.0 

16.1 

15.2 

14.8 

420 

19.0 

19.5  20.0 

20.3 

20.3 

20.3 

20.1 

19.4 

19.0 

18.9 

18.1 

17.3 

16.5 

440 

17.8 

18.7 

19.2 

19.7 

20.1 

20.4 

20.7 

20.7 

20.5 

20.2 

19.8 

19.5 

18.G 

460 

15.9 

16.8  17.6 

18.6 

19.2 

19.9 

20.3 

20.6 

21.0 

20.9 

20.9 

20.8 

20.3 

480 

13.5 

14.6 

15.5 

16.6 

17.7 

18.5 

19.3 

19.9 

20.5 

20.8 

21.1 

21.2 

21.2 

500 

11.3 

12.4 

13.4 

14.4 

15.5 

15.5 

17.7 

18.6 

19.1 

19.9 

20.7 

21.0 

21.4 

520 

9.3 

10.2 

11.2 

12.2 

13.3 

14.2 

15.4 

16.4 

17.6 

18.4 

19.2 

19.8 

20.6 

540 

7.9 

8.6 

9.4 

10.1 

11.1 

12.1 

13.1 

14.2 

15.3 

16.3 

17.4 

18.3 

19.2 

560 

6.9 

7.2 

7.8 

8.4 

9.2 

10.1 

11.0 

11.9 

13.1 

14.1 

15.2 

16.2 

17.2 

580 

6.0 

6.3 

6.6 

7.0 

7.6 

8.4 

9.1 

9.9 

10.9 

11.9 

12.9 

14.1 

15.0 

600 

5.6 

5.6 

5.8 

6.1 

6.5 

6.8 

7.4 

8.1 

*8.8 

9.9 

10.7 

11.8 

12.8 

620 

5.2 

5.4 

5.3 

5.3 

5.5 

5.9 

6.3 

6.6 

7.2 

8.0 

8.7 

9.5 

10.6 

640 

5.9 

5.6 

5.2 

4.9 

5.0  5.0 

5.2 

5.5 

5.8 

6.4 

7.0 

7.6 

8.5 

660 

6.9 

6.3 

5.7 

5.4 

5.0  4.8 

4.5 

4.7 

4.9 

5.1 

5.5 

6.0 

6.8 

680 

8.6 

7.6 

6.9 

6.2 

5.6  5.1 

4.8 

4.6 

4.2 

4.2 

4.5 

4.6 

5.l| 

700 

11.0 

10.0 

P.7 

7.8 

6.8 

6.3 

5.6 

5.0 

4.6 

4.2 

4.2 

4.0 

4.2 

720 

13.9 

12.5 

11.2 

10.3 

9.1 

7.9 

7.1 

6.2 

5.6 

4.8 

4.5 

4.2 

3.8 

740 

16.8 

15.5 

14.4 

13.0 

11.7110.5 

9.4 

8.4 

7.2 

6.5 

5.6 

5.0 

43 

760 

19.7 

185 

17.2 

15.9 

14.7  '  13.5 

12.2 

10.8 

9.8 

8.9 

7.6 

6.7 

5.9 

780 

22.2 

21.2 

20.1 

19.0 

17.6 

16.3 

15.1 

14.0 

12.6 

11.6 

10.2 

9.2 

8.1 

800 

24.4 

23.4 

22.2 

21.3 

20.3 

19.2 

18.0 

16.7 

15.4 

14.3 

13.2 

11.9 

10.8 

820 

25.9 

25.1 

24.4 

23.3 

22.3 

21.6 

20.4 

19.4 

18.2 

17.2 

15.9 

14,6 

13.6 

840 

27.2 

26.6 

25.8 

25.0 

24.3 

23.5 

22.4 

21.6120.5 

19.4 

18.4 

17.3 

16.4 

860 

27.5 

27.1 

26.8 

26.4 

25.5 

24.8 

24.3 

23.3  22.2 

21.5 

20.5 

19.6 

18.4 

880 

27.7 

27.5 

27.2 

27.0 

26.5 

26.0 

25.5 

24.7  24.1 

23.2  22.0 

21.4 

20.4 

900 

27.6 

27.8 

27.9 

27.6 

27.1 

26.7 

26.5125.7  25.3 

24.6  23.9 

23.0 

22.0 

I 

920 

27.3 

27.5 

27.5 

27.6 

27.7 

27.5 

27.2 

26.7  26.3 

25.7  '25.1 

24.3  23.6 

940 

26.1 

26.7 

27.2 

27.4 

27.7 

27.7 

27.6 

27.5  27.1 

26.  6  ''  26.2 

25.6 

25.5 

960 

24.7 

25.4 

26.2 

88  J 

27.2 

27.5 

27.7 

27.7  27.6 

27.4127.1 

27.0 

26.2 

980 

22.6 

23.7 

24.6 

25.3 

25.9 

26.8 

27.2 

27.5  27.7 

27.8  27.6 

27.5 

27.1 

1000 

20.1 

21.1 

22.2 

23.4 

24.3 

25.2 

25.8 

26.6  27.2 

27.6  (  27.7 

27.6 

27.6 

240 

250  1  260 

270 

280 

290 

300 

310  320 

330  340 

350 

360 

TABLE  XXX. 


Perturbations  produced  by   Venus. 

Arguments  II.  and  III. 

III. 


II. 

360 

370 

380 

390 

400 

410 

420 

430 

440 

450 

460 

470 

480 

0 

27.6 

27.7 

27.3 

26.7 

26.2 

25.5 

24.7 

23.8 

23.1 

22.3 

21.3 

20.2 

19.3 

20 

27.7 

27.8 

27.8 

27.6 

27.4 

26.8 

26.2 

25.6 

24.8 

24.0 

23.1 

22.0 

20.9 

40 

26.9 

27.3 

27.6 

27.9 

27.9 

27.7 

27.5 

27.1 

26.3 

25.6 

24.9 

24.0 

23.2 

60 

25.3 

26.0 

26.8(27.1 

27.5 

27.9 

27.8 

27.7 

27.3 

27.1 

26.7 

25.9 

25.0 

80 

23.1 

24,0 

25.1 

25.9 

26.5 

27.3 

27.5 

27.9 

28.2 

28.0 

27.6 

27.5 

27.2 

100 

20.8 

21.8 

22.6 

23.6 

24.6 

25.5 

26.2 

26.7 

27.2 

27.5 

27.6 

27.8 

27.4 

120 

18.5 

19.6 

20.6 

21.5 

22.4 

23.2 

24.1 

25.1 

25.8 

26.4 

26.9 

27.3 

27.5 

140 

17.1 

17.9 

18.6 

19.3 

20.3 

21.3 

22.0 

22.9 

23.7 

24.7 

25.5 

26.0 

26.7 

160 

16.5 

17.1 

17.4 

18.1 

18.8 

19.3 

20.1 

21.0 

21.9 

22.6 

23.5 

24.2 

25.1 

180 

16.9 

17.0 

17.1 

17.4 

18.0 

18.4 

18.9 

19.4 

20.1 

20.7 

21.2 

22.2 

23.0 

200 

17.7 

17.5 

17.7 

17.7 

17.6 

18.1 

18.3 

18.7 

19.2 

19.7 

20.1 

20.8 

21.5 

220 

18.3 

18.2 

18.3 

18.3 

18.3 

18.3 

18.6 

18.7 

18.9 

19.3 

19.5 

20.0 

20.4 

240 

18.6 

18.8 

18.9 

18.9 

18.9 

19.0 

19.2 

19.1 

19.2 

19.5 

19.6 

19.7 

19.9 

260 

18.2 

18.5 

18.7 

18.8 

19.0 

19.3 

19.5 

19.6 

19.9 

19.9 

20.0 

20.1 

20.2 

280 

16.8 

17.4 

17.9 

18.3 

18.7 

19.1 

19.3 

19.8 

20.0 

20.2 

20.4 

20.6 

20.8 

300 

15.5 

15.8 

16.2 

16.6 

17.6 

18.1 

18.5 

19.2 

19.4 

19.9 

20.6 

20.8 

20.9 

320 

13.8 

14.2 

14.6 

15.1 

15.6 

16.2 

16.8 

17.7 

18.3 

18.9 

19.5 

20.1 

20.8 

340 

12.6 

12.9 

13.0 

13.3 

13.7 

14.4 

14.9 

15.5 

16.2 

17.1 

18.0 

18.6 

19.4 

360 

12.3 

12.1 

11.9 

12.0 

12.3 

12.5 

13.0 

13.4 

14.2 

14.9 

15.7 

16.5 

17.3 

380 

13.1 

12.5 

11.9 

11.6 

11.5 

11.4 

11.6 

11.7 

12.3 

12.7 

13.3 

14.0 

15.0 

400 

14.8 

13.9 

13.1 

12.5 

11.7 

11.2 

11.1 

10.9 

11.0 

11.1 

11.4 

12.0 

12.6 

420 

16.5 

15.7 

15.1 

14.3 

13.4 

12.5 

11.7 

11.1 

10.8 

10.8 

10.5 

10.6 

10.7 

440 

18.6 

17.9 

17.1 

16.1 

15.6 

14.4 

13.5 

12.8 

11.9 

11.1 

10.6 

10.3 

10.3 

460 

20.3 

19.8 

19.3 

18.5 

17.6 

16.8 

15.9 

14.7 

13.7 

12.9 

12.0 

11.1 

10.9 

480 

21.2 

21.1 

20.8 

20.3 

19.7 

19.1 

18.3 

17.4 

16.4 

15.0 

14.1 

13.2 

12.2 

500 

21.4 

21.4 

21.4 

21.3 

21.1 

20.8 

20.0 

19.5 

18.8 

17.8 

17.0 

15.7 

14.4 

520 

20.6 

21.2 

21.7 

21.7 

21.5 

21.5 

21.4 

21.1 

20.5 

19.8 

19.1 

18.2 

17.6 

540 

19.2 

20.0 

20.7 

21.1 

21.8 

22.0 

21.8 

21.7 

21.5 

21.2 

20.9 

20.3 

19.6 

560 

17.2 

18.4 

19.0 

20.0 

20.8 

21.1 

22.7 

21.9 

22.2 

22.1 

21.9 

21.7 

21.1 

580 

15.0 

16.0 

17.3 

18.2 

19.1 

19.9 

20^.8 

21.1 

21.7 

22.0 

22.2 

22.3 

22.1 

600 

12.8 

13.9 

15.1 

15.9 

17.2 

18.0 

19.0 

19.9 

20.6 

21.3 

21.8 

22.0 

22.4 

620 

10.6 

11.5 

12.7 

13.7 

14.9 

16.0 

17.1 

18.3 

19.1 

19.9 

20.8 

21.3 

22.0 

640 

8.5 

9.5 

10.4 

11.3 

12.3 

13.7 

14.9 

16.0 

17.1 

18.1 

19.0 

19.9 

20.7 

660 

6.8 

7.4 

8.2 

9.1 

10.1 

11.1 

12.2 

13.6 

14.6 

15.8 

17.1 

18.1 

19.0 

680 

5.1 

5.7 

6.4 

7.1 

7.9 

8.7 

9.7 

11.0 

12.1 

13.1 

14.1 

15.7 

16.8 

700 

4.2 

4.4 

4.7 

5.1 

5.8 

6.7 

7.4 

8.4 

9.4 

10.6 

11.5 

13.0 

14.1 

720 

3.8 

3.8 

3.8 

4.0 

4.4 

4.8 

5.4 

5.9 

6.9 

8.0 

9.1 

10.1 

11.5 

740 

4.3 

3.9 

3.8 

3.7 

3.6 

3.8 

3.9 

4.4 

4.9 

5.7 

6.4 

7.4 

8.9 

760 

5.9 

5.1 

4.4 

4.0 

3.6 

3.4 

3.4 

3.5 

3.9 

4.3 

4.7 

5.2 

5.9 

780 

8.1 

7.1 

6.1 

5.3 

4.6 

4.1 

3.7 

3.3 

3.3 

3.1 

3.4 

3.6 

4.1 

800 

10.8 

9.7 

8.5 

7.5 

6.5 

5.6 

4.9 

4.2 

3.8 

3.4 

3.2 

3.1 

3.1 

820 

13.6 

12.5 

11.2 

10.1 

9.0 

8.0 

6.9 

6.1 

5.3 

4.7 

3.9 

3.7 

3.1 

840 

16.4 

15.1 

1C.  7 

12.9 

11.7 

10.6 

9.5 

8.6 

7.5 

6.6 

5.7 

4.9 

4.4 

860 

18.4 

17.5 

16.6 

15.4 

14.3 

13.1 

12.1 

1.1 

0.0 

9.1 

7.9 

7.0 

6.3 

880 

20.4 

19.6 

18.7 

17.5 

16.6 

15.6 

14.5 

3.6 

2.5 

11.5 

0.4 

9.5 

8.6 

900 

22.0 

21.1 

20.2 

19.4 

18.7 

17.7 

16.5 

5.7  i 

14.7 

13.8 

2.5 

1.9 

109 

920 

23.6 

22.7 

21.7 

21.1 

20.1 

19.4 

18.4 

7.5 

16.7 

15.6 

4.8 

3.9 

13.1 

940 

25.5 

24.1 

23.4 

22.4 

21.4 

20.6 

19.9 

9.0  18.2  17.3 

6.6 

5.7 

14.8 

960 

26.2 

25.6 

24.7 

24.1 

23.3 

22.3 

21.3 

20.6  19.0  18.9 

7.9 

7.1 

16.3 

980 

27.1 

26.7 

26.3 

25.5 

24.9 

23.8 

23.4  2.2  '21.0  20.4 

9.4 

8.6 

17.7 

1000 

27.6 

27.7 

27.3 

26.7 

26.2 

25.5 

24.7  23.8  23.1  22.3 

,L3 

0.2 

19.3 

360 

370 

380 

390 

400  410 

420  !  430  440  450 

460 

470 

480 

TABLE  XXX. 


Perturbations  produced  by  Venus. 
Arguments  II  and  III. 

in. 


II. 

480 

490 

500 

510 

520 

530 

540 

550 

560 

570 

580 

690  600 

0 

19.3 

18.3 

17.4 

16.6 

15.7 

15.0 

14.2 

13.6 

13.1 

12.3 

11.7 

11.3  10.8 

20 

20.9 

20.2 

19.  1 

18.2 

17.1 

16.2 

15.5 

14.7 

14.1 

13.3 

12.7 

12.2  11.5 

40 

23.2  22.0 

20.8 

20.1 

18.9 

17.9 

17.1 

15.9 

15.1 

14.4 

13.7 

13.0  12.3 

60 

25.0  |  24.0 

23.2 

22.0 

20.7 

19.9 

18.9 

17.7 

16.8 

15.8 

14.9 

14.0  13.3 

80 

27.2 

26.4 

25.6 

24.1 

23.2 

22.1 

20.8 

20.0 

18.7 

17.9 

16.6 

15.6  !  14.8 

100 

27.4 

27.2 

26.8 

26.3. 

25.4 

24.5 

23.5 

22.2 

20.9 

20.0 

18.6 

17.6 

16.6 

120 

27.5 

27.5 

27.6 

27.1 

26.8 

26.3 

25.4 

24.6 

23.7 

22.4 

21.0 

20.1 

18.8 

140 

26.7 

27.0 

27.2  27.4 

27.3 

27.4  26.9 

26.2 

25.4 

24.6 

23.9 

22.6  '21.1 

160 

25.1 

25.6 

26.1  26.7 

26.9 

27.3  j  27.1 

27.0 

26.9 

26.4 

25.5 

24.7 

23.9 

180 

23.0 

23.8 

24.5  ;  25.0 

25.7 

26.3  26.7 

26.8 

27.0 

26.8 

26.6 

26.2 

25.6 

200 

21.5 

22.2 

22.8  !  23.5 

24.1 

24.7 

25.5 

25.8 

26.3 

26.6 

26.6 

26.6 

26.4 

220 

20.4 

21.0 

21.5 

22.0 

22.6 

23.2 

23.8 

24.5 

25.0 

25.4 

25.8 

26.0 

26.2 

240 

19.9 

20.4 

20.8 

21.2 

21.6 

21.8 

22.2 

22.6 

231 

23.3 

23.9 

24.2 

24.6 

260 

20.2 

20.3 

20.6 

21.2 

21.4 

21.7 

21.9 

22.2 

223 

22.7 

23.1 

23.3 

23.6 

280 

20.8 

20.8 

21.0 

21.1 

21.3 

21.4 

21.5 

21.8 

22.0 

22.2 

22.7 

23.0 

23.3 

300 

20.9 

21.0 

21.5 

21.7 

21.7 

22.0 

22.0 

22.1 

22  1 

22.2 

224 

22.6 

22.8 

320 

20.8 

21.2 

21.5 

21.6 

22.0 

22.3 

22.5 

22.5 

226 

22.7 

22.8 

22.8 

22.9 

340 

19.4 

20.2 

20.8 

21.5 

21.9 

22.1 

22.6 

23.0 

23.2 

23.4 

23.3 

23.4 

23.5 

360 

17.3 

18.4 

19.5 

20.0 

20.6 

21.5 

22.2 

22.7 

23.0 

23.7 

23.7 

24.0 

24.2 

380 

15.0 

15.9- 

16.9 

17.8 

18.6 

19.6 

20.6 

21.5 

22.3 

22.9 

235 

23.9 

24.5 

400 

12.6 

13.2 

14.2 

15.4 

16.2 

17.3 

18.1 

19.2 

20.3 

21.4 

224 

23.0 

23.7 

420 

10.7 

11.2 

12.0 

12.5 

13.5 

14.5 

15.6 

16.7 

17.7 

18.7 

201 

21.0 

22.0 

440 

10.3 

10.2 

10.3 

10.5 

11.3 

12.0 

12.9 

13.6 

14.7 

16.0 

17.0 

18.3 

19.5 

460 

10.9 

10.1 

9.9 

9.9 

9.9 

10.1 

10.7 

11.3 

12.2 

13.0 

140 

15.1 

16.5 

480 

12.2 

11.4 

10.7 

10.1 

9.7 

9.5 

9.7 

9.9 

10.2 

10.7 

11.7 

12.5 

13.4 

500 

14.4 

13.6 

12.5 

11.6 

10.9 

10.2 

9.8 

9.4 

9.3 

9.6 

9.8 

10.2 

11.1 

520 

17.6 

16.2 

15.1 

13.9 

12.9 

11.9 

10.9 

10.3 

9.8 

9.5 

9.2 

9.2 

9.6 

540 

19.6 

18.6 

18.0 

16.7 

15.4 

14.5 

12.2 

12.3 

11.3 

10.5 

10.1 

9.5 

9.3 

560 

21.1 

20.4 

19.8 

19.0 

18.2 

17.2 

16.0 

14.8 

13.7 

12.7 

11.7 

10.9 

10.2 

580 

22.1 

21.8 

21.5 

20.9 

20.3 

19.3 

18.6 

17.3 

16.5 

15.4 

14.0 

129 

12.2 

600 

22.4 

22.4 

22.2 

22.2 

21.5 

21.2 

206 

19.5 

19.1 

17.7 

16.8 

15.8 

14.4 

620 

22.0 

22.3 

22.4 

22.4 

22.3 

22.3 

21  9 

21.5 

20.9 

20.0 

19.3 

18.0 

16.9 

640 

20.7 

21.7 

22.0 

22.3 

22.6 

22.5 

226 

22.4 

22.0. 

21.6 

21.1 

203 

19.6 

660 

19.0 

20.0 

20.8 

21.3 

22.1 

22.3 

226 

22.8 

22.7 

22.6 

22.2 

21.8 

21.3 

680 

16.8 

18.0 

19.0 

19.9 

20.8 

21.5 

22  1 

22.6 

22.7 

23.0 

23.0 

22.8 

22.4 

700 

14.1 

15.2 

16.8 

17.9 

18.8 

20.0 

221 

21.5 

22.2 

22.6 

22.9 

230 

23.2 

720 

11.5 

12.7 

13.9 

15.0 

16.4 

17.9 

18.6 

19.7 

20.8 

21.6 

22.3 

227 

23.0 

740 

8.9 

9.8 

0.9 

12.2 

13.6 

14.8 

16.2 

17.5 

18.7 

19.5 

20.6 

21.6 

22.3 

760 

5.9 

6.8 

8.0 

9.3 

10.3 

11.8 

13.2 

14.5 

15.9 

17.4 

18.2 

19.5 

20.5 

780 

4.1 

4.9 

5.6 

6.4 

7.5 

8.6 

9.9 

11.  1 

12.6 

14.0 

15.6 

16.8 

18.1 

800 

3.1 

3.3 

4.4 

4.8 

5.5 

6.1 

6.9 

7.9 

9.4 

10.7 

12.1 

13.4 

14.9 

820 

3.1 

3.1 

3.2 

3.1 

3.6 

3.9 

4.8 

5.7 

65 

75 

8.7 

10.0 

11.5 

840 

4.4 

3.7 

3.5 

3.2 

3.2 

3.1 

3.4 

3.7 

4.1 

5.0 

6.2 

7.0 

8.2 

860 

6.3 

5.5 

4.6 

4.1 

3.6 

3.4 

3.3 

3.2 

3.4 

3.4 

4.0 

4.5 

5.6 

880 

8.6 

7.6 

6.7 

5.9 

5.2 

4.5 

4.1 

3.8 

3.5 

34 

3.4 

3.6 

3.9 

900 

10.9 

10.0 

9.1 

8.3 

7.2 

6.5 

5.8 

5.1 

4.4 

42 

3.8 

3.6 

3.6 

920 

13.1 

12.1 

1.2 

10.3 

9.6 

8.7 

7.7 

6.9 

6.3 

5.8 

5.1 

4.6 

4.2 

940 

14.8 

14.1 

13.1 

12.4 

11.5 

10.8 

9.8 

9.1 

8.3- 

7.6 

6.8 

6.5 

5.9 

960 

16.3 

15.4 

14.6 

14.0 

13.2 

12.6;  11.7 

1.0 

0.1 

9.6 

8.8 

8.1 

7.5 

980 

17.7 

16.8 

16.2 

15.2 

14.5 

13.9  13.1 

2.5 

1.8 

11.2 

10.5 

9.7 

9.3 

1000 

19.3 

18.3 

17.4 

16.6 

15.7 

15.0  14.2 

3.6 

3.1 

12.3 

11.7 

1.3 

0.8 

# 

480 

490 

500 

510 

520 

530 

540 

550 

560 

570 

580 

500 

000 

TABLE  XXX. 

Perturbations  produced  by  Venus. 

Arguments  II.  and  III. 

111. 


27 


11. 

600  610 

620 

630 

640 

650 

660 

670  680  690 

700 

710 

720 

0 

10.8 

10.2 

9.5 

9.i 

8.4 

7.9 

7.4 

7.0  6.6  6.3 

5.9 

5.5 

5.4 

20 

11.5 

11.3 

10.7 

10.4 

9.8 

9.4 

8.9 

8.5  7.9  7.7 

7.3 

6.7 

6.6 

40  j  12.3 

12.0 

11.5 

11.0 

10.7 

10.3 

10.0 

9.6  9.3  8.9 

8.5 

8.1 

7.8 

60  1  13.3 

12.7  12.1 

11.6 

11.2 

10.9 

10.5 

10.2  10.0  !  9.8 

9.5 

9.2 

8.9 

80  14.8 

13.6 

12.9 

12.4 

11.8 

11.3 

10.9 

10.7  10.3  !  9.9 

9.8 

9.8 

9.6 

100 

16.6 

15.4 

14.4 

13.4 

12.6 

12.1 

11.5 

11.0  10.6 

10.2 

10.0 

9.9 

9.6 

120 

18.8 

17.7 

16.4 

15.3 

14.3 

13.2 

12.4 

11.6  11.2 

10.6 

10.1 

10.1 

9.6 

140 

21.1 

20.1 

18.9 

17.7 

16.5 

15.2 

14.2 

13.0  12.3 

11.6 

11.  1 

10.3 

9.9 

160 

23.9 

22.9 

21.5 

20.4 

19.2 

17.9 

16.6 

15.3  14.1 

13.1 

12.0 

11.2 

10.5 

180 

25.6 

24.8 

23.9 

22.9 

21.6 

20.6 

19.1 

18.0  16.7 

15.5 

14.3 

12.9 

12.0 

200 

26.4 

26.0 

25.6 

24.9 

24.0 

22.9 

21.7 

20.8  ,  19.3 

18.1 

16.9 

15.5 

14.4 

220 

26.2 

26.3 

26.1 

25.8 

25.3 

24.9 

24.1 

23.1  21.2 

20.9 

19.7 

18.3 

17.1 

240 

24.6 

25.1 

25.1 

25.3 

25.2 

25.1 

24.7 

24.3  24.0 

23.0 

21.9 

21.3 

20.2 

260 

23.6 

23.9 

24.2 

24.5 

24.7 

24.8 

24.9 

24.6  24.3 

23.8 

23.4 

22.9 

21.6 

280 

23.3 

23.6 

23.9 

24.2 

24.7 

24.8 

25.0 

24.9  24.9 

24.8 

24.4 

24.0 

23.5 

,  300 

22.8 

23,0 

23.3 

23.4 

23.8 

24.0 

24.1 

24.5 

24.5 

24.6 

24.5 

24.4 

24.0 

320 

22.9 

23.0 

23.1 

23.2 

23.4 

23.3 

23.6 

23.8 

24.0 

23.9 

24.2 

24.2 

24.2 

340 

23.5 

23.5 

23.5 

23.4 

23.5 

23.6 

236 

23.5 

23.5 

23.6 

23.9 

23.8 

23.8 

360 

24.2 

24.2 

24.3 

24.2 

24.2 

24.0 

23.7 

23.9 

24.0 

23.7 

23.7 

23.6 

23.6 

380 

24.5 

24.6 

24.8 

25.1 

24.8 

24.9 

25.0 

24.9 

24.6 

24.5 

24.5 

24.3 

24.0 

400 

23.7 

24.3 

24.7 

25.0 

25.4 

25.7 

25.7 

25.5 

25.5 

25.4 

25.2 

24.8 

24.6 

420 

22.0 

23.0 

23.7 

24.6 

25.0 

25.7 

26.1 

26.2 

26.3 

26.5 

26.2 

26.0 

25.9 

440 

19.5 

20.8 

21.7 

22.7 

23.7 

24.6 

25.4 

26.0 

26.5 

26.7 

26.9 

27.0 

26.9 

460 

16.5 

17.8 

19.0 

20.1 

21.4 

22.3 

23.5 

24.8 

25.4 

26.1 

26.7 

27.1 

27.3 

480 

13.4 

14.5 

15.6 

17.0 

18.5 

19.7 

20.9 

22.1 

23.2 

24.4 

25.4 

26.2 

26.8 

500 

11.1 

12.0 

13.0 

13.8 

14.9 

16.3 

17.9 

19.1 

20.5 

21.6 

22.9 

24.2 

25.1 

520 

9.6 

9.8 

10.5 

11,5 

12.4 

13.4 

14.4 

15.5 

17.1 

18.4 

19.9 

21.2 

22.3 

540 

9.3 

9.0 

9.2 

9.6 

10.3 

11.0 

11.9 

12.8 

13.9 

15.1 

16.5 

17.9 

19.4 

560 

10.2 

9.7 

9.3 

9.1 

9.1 

9.4 

10.0 

10.6 

11.5 

12.4 

13.3 

14.5 

16.0 

580 

12.2 

11.3 

10.4 

9.9 

9.4 

9.0 

9.2 

9.3 

9.7 

10.4 

11.0 

12.0 

12.7 

600 

14.4 

13.3 

12.5 

11.6 

10.8 

10.1 

9.6 

9.4 

9.1 

9.3 

9.9 

10.0 

10.8 

620 

16.9 

16.1 

14.9 

13.7 

12.7 

12.0 

11.1 

10.4 

9.8 

9.5 

9.5 

9.3 

9.7 

640 

19.6 

18.4 

17.4 

16.3 

15.2 

14.2 

13.1 

12.1 

11.3 

10.6 

10.1 

9.6 

9.5 

660 

21.3 

20.6 

19.9 

18.7 

17.8 

16.7 

15.6 

14.4 

13.4 

12.4 

11.7 

11.0 

10.2 

680 

22.4 

22.0 

21.5 

20.8 

20.2 

19.0 

18.1 

17.0 

15.8 

14.7 

13.7 

12.8 

12.0 

700 

23.2 

23.2 

22.6 

22.2 

21.7 

21.0 

20.5 

19.3 

18.3 

17.3 

16.0 

15.0 

14.1 

720 

23.0 

23.3 

232 

234 

23.1 

224 

21  9 

21  3 

9,08 

19.5 

18.5 

176 

164 

740 

22.3 

22.8 

23.2 

23.4 

23.6 

23.6 

23.3 

22.8 

22.2 

21.6 

21.1 

19.9 

18.8 

760 

20.5 

21.4 

22.5 

22.8 

23.3 

23.7 

23.6 

23.8 

23.5 

23.3 

22.7 

21.8 

21.3 

780 

18.1 

19.2 

20.4 

21.3 

22.3 

23.0 

23.3 

23.7 

23.8 

24.0 

23.8 

23.5 

23.0 

800 

14.9 

16.4 

17.7 

19.1 

20.1 

21.2 

21.1 

22.9 

23.4 

23.8 

24.1 

24.2 

23.9 

820 

11.5 

12.9 

14.3 

15.8 

17.8 

18.7 

20.0 

20.9 

22.0 

22.7 

23.5 

23.9 

24.0 

840 

8.2 

9.5 

10.8 

12.2 

13.8 

15.2 

16.6 

18.1 

19.5,20.6 

21.7 

22.6 

23.3 

860 

5.6 

6.8 

7.7 

8.8 

10.2 

11.5 

13.2 

14.7 

16.0  '  17.4 

19.0 

20.2 

21.3 

880 

3.9 

4.4 

5.2 

6.1 

7.2 

8.2 

9.7 

10.9 

12.5 

14.1 

15.4 

16.8 

18.2 

900 

3.6 

3.6 

3.9 

4.2 

5.0 

5.7 

6.6 

7.8 

9.1 

10.3 

11.8 

13.4 

14.8 

920 

4.2 

3.8 

3.9 

3.9 

4.0 

4.3 

4.7 

5.4 

6.4 

7.3 

8.6 

9.8 

11.2 

940 

5.9 

5.1 

4.6 

4.4 

4.2 

4.3 

4.3 

4.3 

4.9 

5.3 

6.3 

7.0 

8.0 

960 

7.5 

6.9 

6.3 

5.8 

5.3 

4.7 

4.7 

4.6 

4.6 

4.6 

4.9 

5.4 

6.0 

980 

9.3 

8.7 

7.9 

7.4 

6.8 

6.4 

6.0 

5.6 

5.2 

5.0 

4.9 

5.1 

5.1 

1000 

10.8 

10.2 

9.5 

9.1 

8.4 

7.9 

7.4 

7.0 

6.6 

6.3 

5.9 

5.5 

5.4 

600 

610 

620 

630 

640 

650 

660 

670 

680  !  690 

700 

710 

720 

28 


TABLE  XXX. 

Perturbations  produced  by   Vtnus. 
Arguments  II.  and  III. 

in- 


11.   720 

730  740 

750 

760 

770 

780 

790  800 

810 

820 

830 

840 

o'  5.4 

5.5   5.8 

6.0 

6.3 

6.8 

7.6 

8.4 

9.3 

10.4 

11.7 

12.9 

14.3 

20   6.6 

6.3   6.0 

6.1 

6.1 

6.2 

6.5 

6.9 

7.7 

8.3 

9.4 

10.2 

11.2 

40   7.8 

7.4 

7.1 

7.0 

6.7 

6.6 

6.8 

6.8 

6.9 

7.2 

7.7 

8.5 

9.3 

60  8.9 

8.8 

8.3 

8.1 

7.8 

7.6 

7.4 

7.4 

7.3 

7.4 

7.4 

7.7 

8.3 

80!  9.6 

9.5 

9.1 

9.11  9.0 

8.8 

8.4 

8.2 

8.1 

8.1 

8.0 

8.1 

8.2 

100  9.6 

9.5 

9.6 

9.5  9.5 

9.3 

9.3 

9.2 

9.2 

9.0 

8.7 

8.7 

8.7 

120  9.6 

9.6 

9.5 

9.3 

9.4   9.6 

9.6 

9.5 

9.5 

9.6 

9.6 

9.6 

9.6 

140  9.9 

9.5 

9.6 

9.4 

9.3!  9.3 

9.0 

9.3 

9.5 

9.8 

9.7 

9.8 

10.0 

160  10.5 

9.9 

9.5 

9.1 

8.9  9.0 

8.9 

9.0 

9.0 

9.0 

9.5 

9.6 

9.9 

180  12.0 

11.0  10.1 

9.7 

9.1   8.8 

8.7 

8.3 

8.5 

8.7 

8.8 

9.0 

9.1 

200  14.4 

13.3  12.0 

11.0 

10.1 

9.4 

8.9 

8.5 

8.2 

8.0 

8.0 

8.3 

8.5 

220  17.1 

15.7 

14.6 

13.2 

12.0 

10.9 

10.2 

9.2 

8.7 

8.3 

7.9 

7.7 

7.7 

240  20.2 

19.1  !  17.8 

16.5 

14.5 

13.4 

12.2 

11.1 

10.0 

9.4 

8.4 

8.0 

7.7 

260  21.6 

21.1  20.1 

19.2 

17.3 

15.9 

14.6 

13.4 

12.4 

11.3 

10.1 

9.1 

8.6 

280  23.5 

22.7  21.6 

21.0 

19.8 

18.8 

17.3 

16.1 

15.0 

13.5 

12.5 

11.5 

10.2 

300  24.0 

23.4  23.2 

22.4  1  21.4 

20.5 

19.8 

18.7 

17.5 

16.1 

15.0 

13.7 

12.4 

320  24.2 

23.9  23.5 

23.1 

22.7 

22.2 

21.2 

20.6 

19.6 

18.6 

17.5 

16.3 

15.1 

340  23.8 

23.9  23.7 

23.5 

23.2 

22.8 

22.3 

21.4 

20.9 

20.5 

19.2 

18.6 

17.4 

360  23.6 

23.6  23.6 

23.3 

23.3 

23.1 

22.9 

22.4 

22.0 

21.4 

20.4 

19.9 

18.9 

380  24.0 

24.0  23.7 

23.5 

23.3 

23.1 

23.1 

22.7 

22.4 

22.2 

21.6 

20.8 

20.0 

400  ,  24.6 

24.4  24.4 

24.0 

23.8 

23.4 

23.2 

23.0 

22.8 

22.4 

22.1 

21.6 

21.3 

420  25.9 

25.6  25.2 

24.8 

24.7 

24.3 

23.9 

23.6 

23.3 

22.9 

22.7 

22.3 

21.7 

440  26.9 

26.6  26.4 

2G.2 

25.9 

25.5 

25.2 

24.9 

24.5 

23.8 

23.4  23.0 

22.8 

460  27.3 

27.6  27.6 

27.4 

27.0 

26.9 

26.5 

28.1 

25.6 

25.0 

24.6 

24.2 

23.7 

480  -26.8 

27.4  27.6 

28.0 

28.1 

28.2 

27.7 

27.4 

27.3 

26.6 

26.2 

25.7 

25.1 

500 

25.1 

26.1  86.8 

27.5 

28.1 

28.2 

28.6 

28.5 

28.4 

28.3 

27.6 

27.2 

26.7 

520 

22.3 

23.9  24.8 

25.9 

26.8 

27.5 

28.1 

28.5 

28.7 

29.0 

28.8 

28.6 

28.4 

540  19.4 

20.7  22.1 

23.4 

24.6 

25.6 

26.5 

27.4 

28.0 

28.7 

28.9 

29.1 

29.2 

560  16.0 

17.3  18.6 

19.9 

21.4 

22.9 

24.1 

25.5 

26.4 

27.3 

28.2 

28.6 

29.2 

580  12.7 

14.1  15.5 

16.8 

18.0 

19.3 

20.9 

22.2 

23.5 

24.9 

26.1 

27.0 

27.8 

600  i  10.8 

11.6  12.7 

13.6 

14.9 

16.2 

17.5 

18.7 

20.2 

21.8 

23.0 

24.4 

25.5 

620!  9.7 

10.0  10.5 

10.7 

12.2 

13.2 

14.4 

15.6 

17.0 

18.3 

19.6 

21.2 

22.6 

640 

9.5 

9.4  9.6 

10.1 

10.4 

11.1 

12.0 

13.0 

14.0 

15.2 

16.5 

17.9 

19.2 

660 

10.2 

10.0  9.7 

9.5 

9.5 

9.9 

10.4 

11.0 

11.7 

12.7 

13.8 

14.9 

16.2 

680 

12.0 

11.2  10.5 

10.0 

9.7 

9.5 

9.6 

10.0 

10.4 

11.0 

11.6 

12.5 

13.8 

700 

14.1 

13.1  ;  12.3 

11.3 

10.7 

10.1 

9.7 

9.7 

9.9 

9.9 

10.4 

10.9 

11.5 

720 

16.4 

15.3 

14.4 

13.3 

12.2 

11.6 

10.9 

10.2 

10.1 

9.9 

10.0 

10.1 

10.4 

740 

18.8 

17.7 

16.7 

15.6 

14.4 

13.5 

12.4 

11.5 

11.1 

10.7 

10.1 

10.0 

10.3 

760  21.3 

20.1 

19.2 

18.1 

16.6 

15.6 

14.7 

13.6 

12.8 

11.9 

11.3 

10.7 

10.3 

780 

230 

22.3 

21.5 

20.5 

19.4 

18.4 

17.2 

l&ft 

14.9 

14.0 

13.0 

12.2 

11.3 

800 

23.9 

23.9 

23.4 

22.6 

21.9 

20.7 

19.8 

18.8 

17.5 

16.2 

15.1 

14.2 

13.4 

820' 

24.0 

24.5 

24.2 

23.9 

23.3 

22.6 

22.3 

21.3 

20.3 

19.4 

18.3 

17.3 

16.2 

840 

23.3 

24.0 

243 

24.5 

24.4 

24.3 

23.8 

23.4 

22.7 

21.7 

20.8 

19.6 

18.3 

860 

21.3 

22.3  23.3 

23.9 

24.2 

24.7 

24.5 

24.5 

24.3 

23.6 

23.1 

21.9 

21.0 

;  880 

18.2 

19.7 

20.9 

22.0 

22.8 

23.8 

24.1 

24.6 

24.8 

24.7 

24.5 

24.0  |  23.5 

900 

14.8 

16.1 

17.6 

19.0 

20.6 

21.5 

22.5 

23.2 

24.1 

24.5 

24.2 

24.8 

24.5 

:  920 

11.2 

12.6 

14.0 

15.5 

17.0 

18.4 

19.9 

21.0 

22.0 

22.9 

23.5 

24.5  24.5 

s  940 

8.0 

9.3 

10.7 

12.0 

13.3 

14  8 

16,4 

17.6 

19.1 

20.4 

21.4 

22.4  23.2 

960 

6.0 

6.9 

7.8 

8.6 

iO.2  '  11.5 

12.7 

14.1 

15.6 

16.9 

18.5 

19.5  20.7 

980 

5.1 

5.5 

6.0 

6.7 

7.7 

8.5 

S.7 

10.9 

12.2 

13.6 

14.8 

16.1 

17.6 

1000 

5.4 

5.5 

6.8 

5.8 

6.3 

6.8 

7.6 

8.4 

9.3 

10.5 

11.7 

12.9 

14.3 

720 

730  740 

750 

760  1  770 

780 

790 

800 

810  820 

830 

840 

TABLE  XXX. 


Perturbations  produced  by  Venus. 

Arguments  II.  and  III. 

III. 


II. 

840 

850 

860 

870  f  880 

890 

900 

910 

920 

930 

940 

950 

960 

0 

14.3 

15.5 

16.9 

18.2 

19.2 

20.2 

21.4 

22.5 

23.0 

23.5 

24.0 

24.2 

24.2 

20 

11.2 

12.4 

13.6 

14.9 

16.2 

17.3 

18.6 

19.6 

20.5 

21.5 

22.4 

23.1 

23.6 

40 

9.3 

10.2 

10.9 

11.8 

13.3 

14.2 

15.5 

16.6 

17.8 

18.8 

19.7 

20.7 

21.6 

60 

8.3 

8.7 

9.5 

10.1 

10.8 

11.6 

12.7 

13.8 

14.9 

15.9 

17.0 

18.1 

19.1 

80 

8.2 

8.3 

8.6 

8.9 

9.6 

10.3 

10.7 

11.6 

12.5 

13.3 

14.5 

15.2 

16.2 

100 

8.7 

8.7 

8.9 

9.0 

9.1 

9.4 

9.9 

10.4 

11.0 

11.7 

12.4 

12.9 

14.0 

120 

9.6 

9.5 

9.3 

9.6 

9.6 

9.7 

9.9 

9.8 

10.4 

10.9 

11.3 

11.8 

12.3 

140 

10.0 

10.2 

10.1 

10.2 

10.1 

10.3 

10.4 

10.5 

10.5 

10.6 

10.9 

11.4 

11.5 

160 

9.9 

10.0 

10.2 

10.4 

10.6 

11.0 

11.0 

10.9 

11.0 

11.3 

11.3 

11.3 

11.6 

180 

9.1 

9.6 

9.9 

10.1 

10.4 

10.7 

11.0 

11.3 

11.5 

11.7 

11.7 

11.9 

12.2 

200 

8.5 

8.8 

9.1 

9.5 

9.7 

10.0 

10.5 

11.0 

11.2 

11.6 

12.0 

12.2 

12.4 

220 

7.7 

7.7 

8.1 

8.4 

8.8 

9.2 

9.7 

10.1 

10.6 

11.0 

11.4 

11.8 

12.3 

240 

7.7 

7.3 

7.4 

7.4 

7.7 

8.0 

8.4 

9.0 

9.6 

10.0 

10.5 

11.0 

11.5 

260 

8.6 

7.9 

7.4 

7.2 

7.1 

7.1 

7.3 

7.6 

8.1 

8.5 

9.3 

10.0 

10.4 

280 

10.2 

9.2 

8.3 

7.9 

7.4 

7.1 

7.0 

6.9 

7.0 

7.3 

7.7 

8.5 

8.8 

300 

12.4 

11.4 

10.4 

9.3 

8.5 

7.8 

7.4 

6.9 

6.7 

6.8 

6.8 

7.0 

7.5 

320 

15.1 

13.9 

12.5 

11.4* 

10.5 

9.7 

8.6 

7.8 

7.4 

7.0 

6.6 

6.5 

6.7 

340 

17.4 

16.4 

15.2 

13.9 

12.7 

11.6 

10.6 

9.7 

8.7 

8.0 

7.3 

6.8 

6.6 

360 

18.9 

18.1 

174 

16.3 

15.1 

13.8 

12.8 

11.7 

10.6 

9.8 

8.8 

8.0 

7A 

380 

20.0 

19.6 

18.8 

17.7 

16.9 

13.0 

15.1 

13.9 

12.7 

11.8 

10.8 

9.8 

8.9 

400 

21.3 

20.6 

19.6 

19.4 

18.4 

17.6 

16.5 

15.7 

14.8 

13.7 

12.8 

11.8 

10.9 

420 

21.7 

21.1 

20.8 

20.3 

19.3 

18.9 

18.2 

17.2 

16.3 

15.3 

14.5 

13.7 

12.6 

440 

22.8 

22.1 

21.6 

20.8 

20.6 

19.7 

19.0 

18.6 

17.7 

16.6 

15.9 

15.1 

14.2 

460 

23.7 

23.3 

22.7 

22.0 

21.6 

20.9 

20.2 

19.5 

18.5 

18.1 

17.3 

16.7 

15.7 

180 

25.1 

24.4 

23.9 

23.3 

22.8 

22.0 

21.4 

20.9 

20.2 

19.3 

18.3 

17.7 

16.9 

500 

26.7 

26.3 

25.7 

24.9 

24.3 

23.6 

23.0 

22.3 

21.4 

20.7 

20.3 

19.1 

18.1 

520 

28.4  i  27.8 

27.3 

26.8 

26.3 

25.6 

24.7 

23.9 

23.3 

22.6 

21.8 

20.8 

20.1 

540 

29.2 

29.2 

28.9 

28.5 

27.8 

27.4 

26.8 

26.1 

25.3 

24.4 

23.7 

23.0 

22.0 

560 

29.2 

29.3 

29.5 

29.6 

29.3 

29.1 

28.8 

28.0 

27.4 

26.9 

26.1 

25.1 

24.3 

580 

27.8 

28.6 

29.0 

29.4 

29.6 

29.8 

29.8 

29.3 

28.0 

28.7 

27.9 

27.3 

26.6 

600 

25.5 

26.7 

27.6 

28.4 

28.9 

29.2 

29.6 

29.9 

29.9 

29.8 

29.3 

29.0 

28.5 

620 

22.6 

23.8 

25.0 

26.2 

27.1 

27.9 

28.8 

29.3 

29.6 

29.8 

30.1 

29.8 

29.6 

640 

19.2 

20.6 

21.6 

23.3 

24.6 

25.2 

26.6 

27.8 

28.3 

28.9 

29.4 

29.7 

29.9 

660 

16.2 

17.5 

18.8 

20.2 

21.1 

22.9 

24.0 

25.1 

26.2 

27.1 

28.2 

28.8 

29.2 

680 

13.8 

14.7 

15.8 

16.9 

18.4 

19.9 

20.6 

22.3 

23.6 

24.9 

25.8 

26.7 

27.5 

700 

11.5 

12.3 

13.4 

14.6 

15.6 

16.7 

18.0 

19.5 

20.7 

22.0 

23.1 

24.2 

25.1 

720 

10.4 

11.0 

11.4 

12.3 

13.3 

14.3 

15.6 

16.4 

17.7 

19.3 

19.9 

21.6 

22.6 

740 

10.3 

10.4 

10.5 

11.0 

11.4 

12.2 

13.3 

14.2 

15.3 

16.5 

17.4 

18.8 

19.5 

760 

10.3 

10.0 

10.2 

10.3 

10.7 

11.0 

11.5 

12.2 

13.1 

14.2 

15.1 

16.0 

17.3 

780 

11.3 

10.8 

10.6 

10.2 

10.2 

10.5 

10.7 

11.1 

11.5 

12.3 

13.2 

14.0 

15.0 

800 

13.4 

12.5 

11.7 

11.0 

10.6 

10.3 

10.3 

10.4 

10.7 

11.0 

11.6 

11.3 

12.2 

820 

16.2 

15.2 

14.4 

13.5 

13.5 

11.9 

11.4 

11.0 

10.9 

10.8 

10.8 

11.2 

11.4 

840 

18.3 

17.1 

16.2 

14.9 

14.1 

13.0 

12.4 

11.7 

11.2 

10.7 

10.6 

11.1 

11.2 

860 

21.0 

20.2 

18.7 

17.7 

16.6 

15.4 

14.3 

13.3 

12.5 

11.9 

11.4 

11.0 

10.9 

880 

23.5 

22.4 

21.3 

20.4 

19.3 

18.0 

17.0 

15.9 

14.8 

13.7 

12.8 

12.0 

12.6 

900 

24.5 

24.2 

23.8 

22.7 

21.9 

19.9 

19.7 

18.6 

17.2 

16.4 

15.3 

14.1 

13.3 

920 

24.5 

24.8 

24.7 

243 

24.1 

23.2 

22.3 

21.3 

20.0 

19.3 

18.0 

16.7 

15.7 

940 

23.2 

24.0 

24.5 

24.6 

24.5  24.5 

24.2 

23.5 

22.7 

21.8 

20.6 

19.5 

18.4 

960 

20.7 

21.9 

22.8 

23.6 

24.0  24.5 

24.5 

24.2 

24.3 

23.7 

22.9 

22.1 

21.0 

980 

17.6 

18.7 

20.1 

21.2 

22.2  23.1 

23.6 

24.0 

24.3 

24.3 

24.3 

23.7 

23.0 

1000 

14.3 

15.5 

16.9 

18.2 

19.2  20.2 

21.4 

22.5 

23.0 

23.5 

24.0 

24.2 

24.2 

840 

850 

860 

870 

880  890 

900 

910 

920 

930 

940 

950 

960 

30 


TABLE  XXX.    XXXI. 


Perturbations  by   Venus. 

Arguments  II  and  III. 

HI. 


Perturbations  by  Mars. 

Arguments  II  and  IV. 

IV. 


11. 

960  | 

970 

980 

990 

1000 

0 

10 

20 

30 

40 

50 

60 

70 

0 

24.2 

23.7 

23.1 

22.5 

21.6 

9.5 

10.2 

10.8 

11.2 

11.5 

11.7 

11.8 

11.5 

20 

23.6 

23.7 

24.0 

23.4 

23.1 

8.3 

9.1 

9.8 

10.5 

10.9 

11.2 

11.5 

11.6 

40 

21.6 

22.4 

22.9 

23.5 

23.5 

7.1 

7.9 

8.8 

9.4 

10.0 

10.6 

10.8 

11.2 

60 

19.1 

20.1 

20.7 

21.5 

22.2 

5.8 

6.7 

7.6 

8.4 

9.1 

9.8 

10.3 

10.5 

80 

16.2 

17.3 

18.4 

19.7 

20.0! 

4.3 

5.3 

6.4 

7.2 

8.0 

8.9 

9.3 

9.9 

100 

14.0 

14.8 

15.6 

16.5 

17.6 

3.3 

4.2 

5.0 

5.9 

6.8 

7.6. 

8.4 

9.1 

120 

12.3 

12.9 

13.7 

14.3 

15.3 

2.4 

3.1 

3.9 

4.8 

5.6 

6.4 

7.3 

8.0 

140 

11.5 

12.0 

12.6 

12.8 

13.6 

2.1 

2.4 

2.9 

3.8 

4.6 

5.5 

6.3 

7.0 

160 

11.6 

11.8 

12.1 

12.3 

12.7 

2.0 

2.2 

2.4 

2.7 

3.5 

4.4 

5.1 

5.9 

180 

12.2 

12.2 

12.3 

12.5 

12.7 

1.9 

2.0 

2.3 

2.6 

2.9 

3.4 

3.9 

4.9 

200 

12.4 

12.7 

12.8 

13.1 

13.2 

2.3 

2.2 

2.2 

2.4 

2.7 

3.0 

3.4 

3.8 

220 

12.3 

12.7 

13.0 

13.3 

13.5 

3.0 

2.6 

2.5 

2.4 

2.5 

2.7 

3.1 

3.5 

240 

11.5 

12.1 

12.4 

13.1 

13.6 

3.7 

3.3 

3.0 

2.9 

2.7 

2.8 

2.9 

3.2 

260 

10.4 

11.0 

11.5 

12.2 

12.8 

4.8 

4.1 

3.7 

3.5 

3.1 

3.1 

3.0 

3.1 

280 

8.8 

9.6 

10.4 

10.7 

11.5 

5.5 

5.1 

4.6 

4.1 

3.8 

3.5 

3.5 

3.4 

300' 

7.5 

7.9 

8.6 

9.0 

10.1 

6.2 

5.8 

5.6 

5.0 

4.8 

4.2 

3.9 

3.8 

320 

6.7 

6.8 

7.3 

7.8 

8.3 

6.9 

6.6 

6.1 

5.9 

5.4 

5.1 

4.7 

4.3 

340 

6.6 

6.4 

6.6 

6.7 

6.2 

7.2 

7.1 

6.9 

6.5 

6.2 

5.8 

5.5 

5.1 

360 

7.4 

6.9 

6.5 

6.5 

6.5 

7.5 

7.4 

7.1 

7.0 

6.8 

6.4 

6.2 

5.8 

880 

8.9 

8.2 

7.5 

6.9 

6.8 

7.5 

7.6 

7.3 

7.3 

7.2 

7.1 

6.7 

6.5 

400 

10.9 

10.0 

9.0 

8.3 

7.5 

7.3 

7.3 

7.5 

7.4 

7.4 

7.4 

7.1 

7.0 

420 

12.6 

11.6 

10.7 

9.9 

9.1 

6.9 

7.0 

7.3 

7.4 

7.4 

7.4 

7.3 

7.5 

440 

14.2 

13.3 

12.5 

11.6 

10.6 

6.5 

6.8 

6.8 

7.1 

7.2 

7.3 

7.3 

7.4 

460 

15.7 

14.8 

13.9 

13.0 

12.1 

6.2 

6.2 

6.5 

6.7 

6.8 

7.1 

7.1 

7.3 

480 

16.9 

16.3 

15.5 

14.5 

13.6 

5.8 

5.9 

6.0 

6.2 

6.4 

6.5 

7.0 

6.9 

500 

18.1 

17.6 

16.6 

15.8 

15.1 

5.3 

5.4 

5.7 

5.8 

6.0 

6.0 

6.3 

6.6 

520 

20.1 

19.2 

18.1 

17.4 

16.5 

5.1 

•6.1 

5.1 

5.3 

5.4 

5.6 

5.8 

6.0 

540 

22.0 

21.0 

20.2 

19.2 

18.1 

4.7 

4.8 

4.8 

4.8 

5.0 

5.1 

5.4 

5.5 

560 

24.3 

23.5 

22.6 

21.5 

20.6 

4.4 

4.5 

4.6 

4.6 

4.7 

4.8 

4.8 

5.0 

580 

26.6 

25.7 

24.9 

23.8 

23.0 

4.2 

4.3 

4.4 

4.3 

4.5 

4.4 

4.4 

4.5 

600 

28.5 

27.8 

27.0 

26.3 

25.4 

4.0 

4.2 

4.3 

4.2 

4.2 

4.2 

4.2 

4.3 

620 

29.6 

29.2 

28.8 

28.2 

27.4 

4.2 

4.0 

4.1 

4.0 

4.0 

4.0 

4,0 

3.9 

640 

29.9 

30.0 

29.9 

29.5 

29.5 

4.3 

4.2 

4.1 

4.0 

4.1 

4.0 

3.9 

3.9 

660 

29.2 

29.5 

29.7 

29.8 

29.9 

4.6 

4.4 

4.3 

4.1 

4.1 

4.1 

4.0 

3.8 

680 

27.5 

28.6 

28.9 

29.2 

29.7 

4.8 

4.6 

4.5 

4.3 

4.2 

4.1 

4.0 

3.9 

700 

25.1 

26.4 

27.3 

27.8 

28.7 

5.3 

5.0 

4.8 

4.5 

4.6 

4.0 

4.1 

4.1 

720 

22.6 

23.9 

25.0 

26.1 

26.8 

5.8 

5.5 

5.1 

5.0 

4.7 

4.5 

4.1 

4.1 

740 

lp 

21.3 

22.5 

23.6 

24.6 

6.5 

6.1 

5.7 

5.4 

5.2 

4.9 

4.6 

4.3 

760 

17.3 

18.6 

19.4 

21.0 

22.1 

7.4 

6.7 

6.4 

6.0 

5.6 

5.3 

5.1 

5.0 

780 

15.0 

15.8 

17.1 

18.5 

19.3 

8.2 

7.6 

6.9 

6.5 

6.4 

5.8 

5.6 

5.3 

800 

12.2 

14.1 

14.8 

15.9 

17.0 

9.2 

8.5 

8.0 

7.3 

6.8 

6.5 

6.1 

5.8 

820 

11.4 

12.0 

12.5 

13.4 

15.4 

10.1 

9.6 

8.8 

8.2 

7.6 

7.1 

6.7 

6.5 

840 

11.2 

11.3 

11.7 

12.2 

13.2 

10.9 

10.4 

9.8 

9.1 

8.4 

7.9 

7.5 

6.9 

860 

10.9 

10.8 

10.9 

11.2 

11.5 

11.7 

11.0 

10.4 

10.0 

9.4 

8.7 

8.2 

7.7 

880 

12.6 

11.3 

11.1 

10.8 

11.0 

12.3 

11.9 

11.3 

10.6 

10.2 

9.7 

8.9 

8.4 

900 

13.3 

12.3 

12.9 

11.3 

11.2 

12.4 

12.2 

11.8 

11.6 

10.8 

10.3 

9.7 

9.3 

920 

15.7 

14.6 

13.7 

12.8 

12.1 

12.3 

12.3 

12.2 

11.9 

11.6 

11.0 

10.5 

9.9 

940 

18.4 

17.3 

16.2 

14.5 

14.0 

12.1 

12.1 

12.2 

12.2 

11.8 

11.4 

11.0 

10.6 

960 

21.0 

20.0 

189 

17.9 

16.7 

11.4 

11.9 

11.9 

12.0 

12.0 

11.7 

11.4 

11.0 

980 

23.0 

22.4 

21.4 

20.3 

19.5 

10.6 

11.1 

11.6 

11.8 

11.9 

11.9 

11.7 

11.4 

1000 

24.2 

23.7 

23.1 

22.5 

21.6 

9.5 

10.2 

10.8 

;  11.2 

11.5 

11.7 

11.8 

11.5 

960 

970 

980 

990 

1000 

0 

10 

20 

30 

40 

50 

60 

70 

TABLE  XXXI. 


31 


Perturbations  produced  by  Mars 

Arguments  II  and  IV. 

IV. 


II. 

70 

80 

90 

100 

110 

120 

130 

140 

150 

160 

170 

180 

190 

200 

0 

11.5 

11.2 

11.0 

10.6 

10.1 

9.9 

9.5 

9.0 

8.6 

8.2 

8.1 

7.8 

7.6 

7.4 

20 

11.6 

11.4 

11.0 

10.9 

10.6 

10.2 

9.7 

9.1 

9.1 

8.8 

8.4 

8.1 

7.9 

7.8 

40 

11.2 

11.3 

11.2 

11.0 

10.8 

10.5 

10.3 

9.8 

9.4 

9.3 

9.1 

8.7 

8.4 

8.2 

60 

10.5 

10.9 

11.1 

10.9 

11.0 

10.9 

10.4 

10.0 

9.7 

9.5 

9.2 

8.8 

8.7 

8.4 

80 

9.9 

10.0 

10.5 

10.9 

10.8 

10.7 

10.4 

10.3 

10.0 

9.7 

9.3 

9.0 

8.8 

8.6 

100 

9.1 

9.5 

9.8 

10.1 

10.6 

10.5 

10.4 

10.3 

10.1 

9.9 

9.6 

9.3 

9.0 

8.8 

120 

8.0 

8.8 

9.3 

9.5 

9.9 

10.2 

10.2 

10.1 

10.0 

9.8 

9.6 

9.4 

9.1 

8.9 

140 

7.0 

7.9 

8.4 

9.0 

9.3 

9.6 

9.9 

9.9 

9.9 

9.7 

9.7 

9.4 

9.3 

8.9 

160 

5.9 

6.5 

7.2 

8.0 

8.5 

8.9 

9.2 

9.6 

9.5 

9.6 

9.5 

9.5 

9.3 

9.1 

180 

4.9 

5.6 

6.4 

6.9 

7.7 

8.3 

8.6 

8.9 

9.4 

9.3 

9.3 

9.3 

9.2 

9.1 

200 

3.8 

4.6 

5.3 

6.0 

6.7 

7.4 

7.9 

8.3 

8.0 

8.9 

9.1 

9.0 

9.0 

8.9 

220 

3.5 

3.9 

4.4 

5.1 

5.8 

6.4 

7.1 

7.6 

7.9 

8.4 

8.6 

8.8 

8.8 

8.7 

240 

3.2 

3.6 

4.0 

4.4 

5.0 

5.5 

6.2 

6.8 

7.4 

7.6 

8.1 

8.4 

8.4 

8.5 

260 

3.1 

3.2 

3.8 

4.1 

4.5 

4.9 

5.4 

5.9 

6.6 

7.1 

7.5 

7.7 

8.0 

8.2 

280 

3.4 

3.4 

3.5 

3.8 

4.2 

4.5 

4.9 

5.5 

5.6 

6.2 

6.8 

7.1 

7.5 

7.8 

300 

3.8 

3.7 

3.7 

3.7 

3.9 

4.4 

4.7 

4.9 

5.4 

5.7 

6.0 

6.6 

6.9 

7.3 

320 

4.3 

4.2 

4.1 

4.0 

4.1 

4.2 

4.4 

4.7 

5.0 

5.4 

5.8 

6.0 

6.4 

6.6 

340 

5.1 

4.9 

4.6 

4.4 

4.4 

4.3 

4.5 

4.5 

5.0 

5.2 

5.5 

5.8 

6.0 

6.3 

360 

5.8 

5.6 

5.3 

5.0 

4.8 

4.8 

4.7 

4.8 

4.9 

5.1 

5.4 

5.5 

5.9 

6.1 

380 

6.5 

6.4 

5.9 

5.7 

5.5 

5.4 

5.1 

5.1 

5.1 

5.1 

5.4 

5.5 

5.7 

5.8 

400 

7.0 

6.7 

6.7 

6.3 

6.1 

5.9 

5.7 

5.6 

5.5 

5.5 

5.5 

5.6 

5.7 

5.9 

420 

7.4 

7.2 

6.9 

7.1 

6.7 

6.4 

6.3 

6.1 

6.0 

5.9 

5.9 

5.8 

5.8 

6.1 

440 

7.5 

7.4 

7.4 

7.0 

7.1 

7.4 

6.8 

6.7 

6.5 

6.3 

6.3 

6.4 

6.2 

6.3 

460 

7.3 

7.4 

7.4 

7.5 

7.4 

7.3 

7.3 

7.2 

7.1 

7.1 

6.7 

6.7 

6.7 

6.7 

480 

6.9 

7.1 

7.3 

7.4 

7.5 

7.3 

7.6 

7.5 

7.4 

7.5 

7.4 

7.2 

7.1 

7.1 

500 

6.G 

6.8 

6.9 

7.2 

7.3 

7.5 

7.5 

7.6 

7.8 

7.7 

7.8 

7.7 

7.6 

7.4 

520 

6.0 

'6.3 

6.5 

6.7 

7.1 

7.2 

7.5 

7.5 

7.7 

7.8 

7.9 

7.6 

7.9 

7.9 

540 

5.5 

5.7 

6.0 

6.3 

6.6 

6.9 

7.1 

7.3 

7.4 

7.7 

7.9 

8.0 

8.2 

8.3 

560 

5.0 

5.2 

5.4 

5.8 

5.9 

6.2 

6.6 

6.9 

7.1 

7.4 

7.7 

7.8 

8.1 

8.2 

580 

4.5 

4.7 

4.9 

5.0 

5.3 

5.7 

6.0 

6.6 

6.8 

7.1 

7.2 

7.5 

7.9 

8.2 

600 

4.3 

4.3 

4.4 

4.6 

4.6 

5.0 

5.3 

5.6 

5.9 

6.5 

6.9 

7.0 

7.4 

7.7 

620 

3.9 

4.0 

4.0 

4.1 

4.3 

4.4 

46 

4.9 

5.3 

5.4 

6.1 

6.6 

6.9 

7.4 

640 

3.9 

3.8 

3.8 

3.8 

3.9 

3.9 

4.1 

4,3 

4.5 

5.0 

5.2 

5.8 

6.3 

6.7 

660 

3.8 

3.7 

3.7 

3.6 

3.6 

3.7 

3.8 

3.9 

4.1 

4.2 

4.5 

5.0 

5.3 

6.0 

680 

3.9 

3.8 

3.6 

3.4 

3.5 

3.4 

35 

3.6 

3.6 

3.7 

3.8 

4.2 

4.6 

4.9 

700 

4.1 

3.9 

3.8 

3.6 

3.5 

3.3 

3.3 

3.2 

3.2 

3.2 

3.5 

3.6 

3.8 

4.2 

720 

4.1 

4.1 

4.0 

3.8 

3.6 

3.5 

3.3 

3.2 

3.3 

3.2 

3.0 

3* 

3.4 

3.6 

740 

4.3 

4.3 

4.2 

4.0 

3.8 

3.7 

3.5 

3.2 

3.0 

3.0 

2.9 

2.8 

2.9 

3.1 

760 

5.0 

4.7 

4.4 

4.3 

4.1 

3.8 

3.7 

3.4 

3.1 

3.0 

2.9 

2.7 

2.7 

2.8 

780 

5.3 

5.1 

4.7 

4.6 

4.4 

4.4 

4.0 

3.8 

3.4 

3.2 

2.9 

2.8 

2.7 

2.5 

800 

5.8 

5.5 

5.4 

4.8 

4.7 

4.7 

4.5 

4.2 

3.9 

3.5 

3.3 

2.9 

2.8 

2.7 

820 

6.5 

6.1 

5.S 

5.6 

5.0 

5.0 

4.9 

4.6 

4.3 

4.1 

3.6 

3.3 

3.0 

2.9 

840 

6.9 

6.7 

6.3 

6.1 

5.8 

5.3 

5.2 

4.9 

4.9 

4.5 

4.2 

3.9 

3.5 

3.1 

860 

7.7 

7.4 

6.9 

6.6 

6.2 

6.2 

5.5 

5.4 

5.2 

5.0 

4.8 

4.4 

4.1 

3.6 

880 

8.4 

7.9 

7.6 

7.1 

6.9 

6.4 

6.4 

5.8 

5.7 

.5.4 

5.2 

5.0 

4.6 

4.3 

900 

9.3 

8.7 

8.3 

7.7 

7.4 

7.1 

C.7 

6.6 

6.1 

6.0 

5.6 

5.4 

5.2 

4.9 

920 

9.9 

9.3 

£.8 

8.4 

7.9 

7.7 

7.3 

6.9 

6.6 

6.3 

6.2 

6.1 

5.6 

54 

940 

10.6 

10.1 

%.5 

8.9 

8.7 

8.2 

7.8 

7.6 

7.2 

7.1 

6.5 

6.5 

6.3 

6.9 

960 

11.0 

10.7 

10.3 

9.7 

9.1 

8.7 

8.4 

8.0 

7.8 

7.4 

7.2 

6.9 

6.7 

6.6 

980 

11.4 

11.0 

10.6 

10.2 

9.8 

9.2 

8.9 

8.4 

8.1 

8.0 

7.6 

7.3 

7.2 

6.9 

1000 

11.5 

112 

11.0 

10.6 

10.0 

9.9 

9.5 

9.0 

8.6 

8.2 

8.1 

7.4 

7.6 

7.4 

70 

80 

90 

100 

110 

120 

130 

140 

150 

160 

170 

180 

190 

200 

32 


TABLE  XXXI. 


Perturbations  produced  by  Mars. 

Arguments  II.  and  IV. 

IV. 


II. 

200 

210 

220 

230 

240 

250 

260 

270 

280 

290 

300 

310 

320 

0 

7.4 

7.2 

7.0 

6.6 

6.4 

6.2 

5.7 

5.3 

4.9 

4.7 

4.1 

3.8 

3.4 

20 

7.8 

7.2 

7.3 

7.2 

7.0 

6.6 

6.3 

6.0 

5.7 

5.3 

5.0 

4.4 

3.9 

40 

8.2 

8.1 

7.6 

7.5 

7.3 

7.2 

6.8 

6.6 

6.2 

5.9 

5.6 

5.2 

4.7 

60 

8.4 

8.0 

7.9 

7.8 

7.6 

7.5 

7.3 

7.1 

6.8 

6.4 

6.1 

5.8 

5.4 

80 

8.6 

8.5 

8.2 

8.0 

7.6 

7.7 

7.6 

7.4 

7.1 

7.0 

6.7 

6.3 

6.0 

100 

8.8 

8.5 

8.6 

8.4 

8.2 

7.6 

7.7 

7.8 

7.6 

7.3 

7.2 

6.9 

6.6 

{  120 

8.9 

8.7 

8.4 

8.4 

8.3 

8.3 

8.0 

7.9 

7.7 

7.6 

7.5 

7.3 

7.0 

:  140 

8.9 

8.7 

8.4 

8.3 

8.2 

8.1 

8.3 

8.0 

7.9 

7.8 

7.7 

7.5 

7.4 

-'  160 

9.1 

8.9 

8.7 

8.4 

8.3 

8.3 

82 

8.1 

8.0 

7.9 

7.9 

7.7 

7.6 

!  180 

9.1 

8.8 

8.7 

8.5 

8.4 

8.2 

8.0 

8.0 

8.1 

7.9 

7.8 

8.0 

7.8 

•  200 

8.9 

8.8 

8.6 

8.4 

8.4 

8.3 

8.1 

8.0 

7.9 

7.8 

7.8 

7.9 

7.9 

220 

8.7 

8.7 

8.6 

8.4 

8.2 

8.1 

8.0 

7.9 

7.8 

7.7 

7.7 

7.3 

7.7 

240 

8.5 

8.4 

8.5 

8.3 

8.1 

8.0 

7.8 

7.8 

7.8 

7.8 

7.8 

7.8 

7.6 

260 

8.2 

8.2 

8.1 

8.1 

8.1 

7.8 

7.8 

7.7 

7.6 

7.6 

7.6 

7.5 

7.4 

280 

7.8 

7.8 

8.0 

7.8 

7.9 

7.9 

7.7 

7.5 

7.5 

7.3 

7.3 

7.4 

7.3 

300 

7.3 

7.6 

7.5 

7.6 

7.7 

7.6 

7.6 

7.6 

•  7.4 

7.3 

7.1 

7.0 

7.1 

320 

6.6 

7.1 

7.3 

7.4 

7.4 

7.3 

7.4 

7.4 

7.3 

7.1 

7.0 

7.0 

6.8 

340 

6.3 

6.4 

6.7 

7.2 

7.1 

7.2 

7.2 

7.1 

7.1 

7.0 

6.9 

6.8 

6.8 

360 

6.1 

6.2 

6.4 

6.5 

6.9 

6.9 

7.0 

7.0 

6.9 

6.8 

6.7 

6.6 

6.5 

380 

5.8 

6.1 

6.3 

6.4 

6.6 

6.7 

6.6 

6.6 

6.7 

6.8 

6.7 

6.6 

6.5 

400 

5.9 

6.0 

6.2 

6.3 

6.4 

6.5 

6.6 

6.6 

6.5 

6.6 

6.6 

6.5 

6.4 

420 

6.1 

6.3 

6.2 

6.4 

6.3 

6.4 

6.5 

6.6 

6.5 

6.5 

6.5 

6.5 

6.4 

440 

6.3 

6.4 

6.4 

6.6 

6.5 

6.6 

6.5 

6.5 

6.5 

6.5 

6.3 

6.3 

6.2 

460 

6.7 

6.5 

6.5 

6.6 

6.7 

6.9 

6.7 

6.6 

6.6 

6.6 

6.5 

6.3 

6.2 

480 

7.1 

7.1 

7.0 

6.9 

6.9 

6.9 

7.0 

7.0 

6.8 

6.7 

6.6 

6.5 

6.3 

500 

7.4 

7.5 

7.4 

7.4 

7.3 

7.2 

7.3 

7.2 

7.1 

6.9 

6.8 

6.8 

6.6 

520 

7.9 

7.8 

7.8 

7.8 

7.8 

7.6 

7.6 

7.5 

7.5 

7.4 

7.1 

7.0 

6.9 

540 

8.3 

8.3 

8.3 

8.2 

8.2 

8.1 

8.0 

7.9 

7.9 

7.8 

7.0 

7.5 

7.2 

560 

8.2 

8.6 

8.4 

8.6 

8.7 

8.5 

8.5 

8.4 

8.2 

8.3 

8.2 

8.0 

7.6 

580 

8.2 

8.3 

8.6 

8.8 

8.8 

9.0 

8.9 

8.9 

8.7 

8.7 

8.6 

8.4 

8.4 

600 

7.7 

8.1 

8.5 

8.6 

8.9 

9.1 

9.1 

9.2 

9.2 

9.1 

9.0 

8.8 

8.7 

620 

7.4 

7.6 

8.0 

8.5 

8.7 

9.0 

9.2 

9.5 

9.5 

9.5 

9.4 

9.3 

9.2 

640 

6.7 

7.2 

7.5 

7.9 

8.3 

8.7 

9.0 

9.3 

9.5 

9.8 

9.8 

9.7 

9.7 

660 

6.0 

6.3 

7.0 

7.3 

7.7 

8.2 

8.7 

9.0 

9.4 

9.7 

9.8 

10.1 

10.0 

680 

4.9 

5.6 

6.0 

6.6 

7.1 

7.7 

8.1 

8.5 

9.0 

9.3 

9.8 

10.0 

10.2 

700 

4.2 

4.5 

5.2 

5.8 

6.4 

6.8 

7.4 

8.0 

8.5 

8.9 

9.2 

9.8 

10.1 

720 

r  3.6 

3.9 

4.3 

4.7 

5.3 

5.9 

6.6 

7.0 

7.8 

8.3 

8.8 

9.1 

9.7 

740 

3.1 

3.3 

3.6 

3.9 

4.4 

4.8 

5.6 

6.2 

6.9 

7.5 

8.0 

8.7 

9.2 

760 

2.8 

2.8 

3.0 

3.3 

3.6 

4.0 

4.4 

5.1 

5.8 

6.5 

7.2 

7.8 

8.4 

780 

2.5 

2.6 

2.5 

2.7 

3.1 

3.3 

3.7 

4.1 

4.8 

5.4 

6.1 

6.9 

7.6 

800 

2.7 

2.5 

2.5 

2.5 

2.5 

2.7 

3.0 

3.4 

3.8 

4.4 

5.0 

5.6 

6.6 

820 

2.9 

2.6 

2.4 

2.3 

2.2 

2.3 

2.6 

2.8 

3.1 

3.4 

4.1 

4.7 

5.4 

840 

3.1 

2.8 

2.6 

2.4 

2.3 

2.2 

2.3 

2.4 

2.6 

2.8 

3.2 

3.8 

4.3 

860 

3.6 

3.3 

3.0 

2.7 

2.4 

2.3 

2.1 

2.2 

2.3 

2.5 

2.7 

3,0 

3.4 

880 

4.3 

3.8 

3.6 

3.2 

2.8 

2.5 

2.3 

2.1 

2.0 

2.2 

2.3 

2.5 

2.6 

900 

4.9 

4.6 

4.2 

3.6 

3.4 

2.9 

2.6 

2.3 

2.2 

2.2 

2.1 

2.2 

2.4 

920 

5.4 

5.1 

4.6 

4.5 

3.9 

3.5 

3.2 

2.9 

2.6 

2.2 

2.0 

2.1 

2.2 

940 

5.9 

5.7 

5.3 

4.9 

4.7 

4.3 

3.8 

3.4 

3.0 

37 

2.4 

2.1 

2.0 

960 

6.5 

6.2 

59 

1.5 

5.1 

4.9 

4.5 

4.0 

3.4 

3.1 

2.8 

2.4 

2.3 

;  980 

6.9 

6.8 

6.4 

6.1 

5.8 

5.4 

5.1 

4.8 

4.3 

3.9 

3.5 

3.0 

2.7 

1000 

7.4 

7.2 

7.0 

6.6 

6.4 

6.2 

5.7 

5.3 

4.9 

4.7 

4.1 

3.8 

34 

200 

210 

220 

230 

240 

250 

260 

270 

280 

290 

300 

310 

320 

TABLE  XXXI. 


33 


Perturbations  produced  by  Mars. 

Arguments  II.  and  IV. 

IV. 


II. 

320 

330  340 

350 

360 

370 

380 

390 

400 

410 

420 

430 

440 

0 

3.4 

2.8 

2.6 

2.4 

2.2 

2.3 

2.3 

2.5 

2.7 

2.9 

3.4 

4.0 

45 

20 

3.9 

3.5   3.1 

2.7 

2.6 

2.4 

2.4 

2.3 

2.5 

•  2.7 

3.0 

3.3 

3.J 

40 

4.7 

4.2  3.9 

3.5 

3.0 

2.8 

2.7 

2.6 

2.5 

2.6 

2.8 

2.9 

3.2 

60 

5.4 

5.0  4.6 

4.2 

3.8 

3.4 

3.1 

2.8 

2.8 

2.7 

2.7 

2.7 

3.0 

80 

6.0 

5.7  5.4 

4.8 

4.4 

4.0 

3.6 

3.4 

3.1 

2.Q 

2.9 

2.9 

2.9 

100 

6.6 

6.3 

5.9 

5.6 

5.2 

4,8 

4.3 

4.0 

3.7 

3.5 

3.2 

3.0 

3.0 

120 

7.0 

6.9 

6.4 

6.1 

5.8 

5.3 

5.2 

4.6 

4.3 

4.0 

3.8 

3.6 

3.4 

140 

7.4 

7.2 

6.9 

6.6 

6.5 

6.1 

5.6 

5.4 

5.0 

4.6 

4.3 

4.0 

3.9 

160 

7.6 

7.5 

7.3 

7.0 

6.8 

6.6 

6.2 

5.9 

5.5 

5.3 

4.9 

4.6 

4.4 

180 

7.8 

7.7 

7.5 

7.4 

7.3 

6.9 

6.7 

6.5 

6.2 

5.8 

5.6 

5.3 

50 

200 

7.9 

7.8  j  7.7 

7.6 

7.5 

7.3 

7.1 

6.9 

6.6 

6.4 

6.1 

5.6 

5.5 

220 

7.7 

7.7 

7.7 

7.8 

7.7 

7.5 

7.3 

7.2 

7.0 

6.7 

6.5 

6.2 

5.9 

240 

7.6 

7.6 

7.6 

7.6 

7.7 

7.6 

7.5 

7.3 

7.2 

7.1 

6.9 

6.6 

6.4 

260 

7.4 

7.3 

7.5 

7.5 

7.5 

7.6 

7.6 

7.5 

7.5 

7.3 

7.1 

7.0 

6.7 

280 

7.3 

7.4 

7.3 

7.3 

.7.4 

7.4 

7.3 

7.4 

7.3 

7.5 

7.2 

7.1 

6.9 

300 

7.1 

7.1 

7.1 

7.0 

7.2 

7.3 

7.3 

7.3 

7.2 

7.2 

7.3 

7.2 

7.1 

320 

6.8 

6.8 

6.9 

6.9 

6.8 

7.0 

7.1 

7.1 

7.1 

'7.1 

7.1 

7.0 

7.2 

340 

6.8 

6.7 

6.6 

6.6 

6.6 

6.8 

6.9 

6.9 

7.0 

7.0  6.9 

6.9 

6.9 

360 

6.5 

6.5 

6.4 

6.3 

6.4 

6.5 

6.6 

6.7 

6.8 

6.8 

6.8 

6.8 

6.9 

380 

6.5 

6.3 

6.3 

6.2 

6.2 

6.2 

6.3 

6.3 

6.4 

6.5 

6.6 

6.7 

6.7 

400 

6.4 

6.2 

6.2 

6.0 

6.1 

6.0 

6.0 

6.0 

6.0 

6.1 

6.2 

6.3 

6.4 

420 

6.4 

6.2 

6.1 

6.0 

5.9 

5.8 

5.9 

5.9 

5.9 

5.9 

5.9 

6.0 

6.0 

440 

6.2 

6.1 

6.0 

5.8 

5.8 

5.7 

5.6 

5.6 

5.6 

5.7 

5.7 

5.8 

5.9 

460 

6.2 

6.0 

5.9 

5.8 

5.7 

5.5 

5.5 

5.4 

5.5 

5.4 

5.5 

5.3 

5.4 

480 

6.3!  6.2 

6.0 

5.7 

5.6 

5.5 

5.4 

5.3 

5.2 

5.2 

5.2 

5.3 

5.3 

500 

6.6 

6.4 

6.2 

6.0 

5.7 

5.4 

5.3 

5.2 

5.1 

5.1 

5.1 

5.0 

5.0 

520 

6.9 

6.7 

6.4 

6.1 

6.1 

5.7 

5.5 

5.1 

5.1 

5.0 

4.9 

5.0 

4.9 

540 

7.2 

7.1 

6.7 

6.5 

6.2 

6.1 

5.8 

5.5 

5.2 

5.0 

4.9 

4.8 

4.8 

560 

7.6 

7.4 

7.3 

7.0 

6.6 

6.3 

6.0 

5.8 

5.4 

5.3 

5.0 

4.7 

4.7 

580 

8-4 

8.0 

7.8 

7.5 

7.0 

6.8 

6.3 

6.2 

5.9 

5.5 

5.3 

5.0 

4.9 

600 

8.7 

8.6 

8.3 

8.0 

7.8 

7.4 

7.0 

6.6 

6.3 

6\0 

56 

5.3 

5.1 

620 

9.2 

9.1 

8.9 

8.6 

8.4 

8.1 

7.6 

7.2 

6.8 

65 

6.1 

5.7 

5.3 

640 

9.7 

9.6 

9.4 

9.p 

9.0 

8.7 

8.2 

7.8 

7.4 

7.0 

6.6 

6.3 

5.8 

660 

10.0 

10,0 

9.9 

9.8 

9.6 

9.3 

8.9 

8.5 

8.2 

7.7 

7.2 

6.8 

6.4 

680 

10.2 

10.4 

10.3 

10.2 

10.1 

9.9 

9.6 

9.3 

9.0 

8.5 

8.1 

7.5 

7.1 

700 

10.1 

10.3 

10.5 

10.6 

10.4 

10.3 

10.1 

9.8 

9.6 

9.3 

8.9 

8.3 

7.8 

720 

9.7 

10.1 

10.3 

10.6 

10.7 

10.6 

10.5 

10.5 

10.2 

10.0 

9.6 

9.2 

8.6 

740 

9.2 

9.6 

10.0 

10.3 

10.6 

10.7 

10.8 

10;  9 

10.6 

]0.5 

10.2 

9.9 

9.4 

760 

8.4 

9.0 

9.5 

9.8 

10.2 

10.6 

10.9 

11.0 

11.0 

11.0 

10.7 

10.5 

10.3 

780 

7.6 

8.2 

8.9 

9.4 

9.9 

10.3 

10.& 

10.9 

11.1 

11.2 

11.0 

10.8 

10.7, 

800 

6.6 

7.3 

7.9 

8.5 

9.2 

9.8 

iai 

10.6 

10.8 

11.1 

11.3 

11.1 

IkO 

820 

5.4 

6.0 

7.0 

7.6 

8.2 

8.9 

9.6 

10.0 

10.5 

10.8 

11.0 

11.3^ 

11.3 

840 

4.3 

5.0 

5.6 

6.5 

7.2 

7.9 

9.8 

9.2 

9.9 

10.3 

107 

10.9 

11.2 

860 

3.4 

4.0 

4.6 

5.3 

6.1 

6.9 

7.5 

8.4 

9.1 

9.6 

10.1 

10.7 

10.9 

880 

2.6 

3.1 

3.7 

4.3 

5.0 

5.7 

66 

7.1 

8.1 

8.7 

9.4 

9.8 

10.4 

900 

2.4 

2.7 

3.0 

3.4 

4.0 

4.6 

5.4 

6.1 

6.9 

7.6 

8.4 

».l 

9.7 

920 

2.2 

2.3 

2.3 

2.8 

3.3 

3.7 

4.3 

5.0 

5.8 

6.5 

7.2 

8.0 

8.7 

040 

2.0 

2.1 

2.3 

2.3 

2.7 

2.9 

3.4 

4.1 

4.7 

5.5 

6.1 

7.0 

7.7 

960 

2.3 

2.2 

2.2 

2.3 

2.3 

2.5 

2.8 

3.2 

3.9 

4.5 

5.1 

5.7 

6.5 

980 

$.7 

2.4 

2.2 

2.3 

2.3 

2.4 

25 

2.8 

3.0 

3.6 

4,1 

4.7 

5.5 

1000 

3.4 

2.8 

2.6 

2.4 

2.2 

2.3 

2.3 

2.5 

2.7 

2.9  3.4 

4.0 

4.0 

i 

320 

330 

340 

3.:0  |  360 

370 

380 

390 

400 

410  1  420 

430 

44D 

34 


TABLE  XXXI. 


Perturbations  produced  by  Mars. 

Arguments  II  and  IV. 

IV. 


n. 

440 

450 

460 

470 

480 

490 

500 

510 

520 

530 

540 

550 

560 

0 

4.5 

5.2 

5.9 

6.6 

7.3 

8.0 

8.5 

9.0 

9.5 

10.0 

10.4 

10.7 

10.9 

20 

3.8 

4.3 

4.9 

5.6 

6.2 

6.9 

7.6 

8.2- 

8.8 

9.3 

9.7 

10.0 

11.4 

40 

3.2 

3.7 

4.2 

4.8 

5.4 

5.9 

6.6 

7.3 

7.9 

8.4 

8.9 

9.4 

9.8 

60 

3.0 

3.2 

3.6 

4.0 

4.5 

5.1 

5.7 

6.3 

6.9 

7.5 

8.0 

8.6 

9.1 

80 

2.9 

3.1 

3.3 

3.5 

3.9 

4.4 

4.9 

5.4 

5.9 

6.5 

7.1 

7.7 

8.2 

100 

3.0 

3.1 

3.2 

3.5 

3.6 

3.8 

4.2 

4.8 

5.3 

5.9 

6.4 

6.9 

7.4 

120 

3.4 

3.3 

3.3 

3.4 

3.5 

3.6 

3.9 

4.2 

4.7 

5.1 

5.6 

6.0 

6.6 

140 

3.9 

3.8 

3.6 

3.6 

3.6 

3.7 

4.0 

4.0 

4.2 

4.6 

5.0 

5.4 

5.9 

160 

4.4 

4.2 

3.9 

4.1 

3.8 

3.7 

4.0 

4.1 

4.2 

4.5 

4.6 

4.9 

5.3 

180 

5.0 

4.8 

4.4 

4.2 

4.2 

4.2 

4.0 

4.1 

4.3 

4.4 

4.4 

4.7 

5.0 

200 

5.5 

5.2 

5.1 

4.8 

4.6 

4.5 

4.5 

4.4 

4.5 

4.5 

4.7 

4.6 

4.8 

220 

5.9 

5.7 

5.5 

5.3 

5.1 

4.9 

4.9 

4.8 

4.7 

4.8 

4.8 

4.9 

5.0 

240 

6.4 

6.2 

5.9 

5.8 

5.6 

5.4 

5.3 

5.2 

5:i 

5.1 

5.1 

5.2 

5.2 

260 

6.7 

6.6 

6.'4 

6.1 

6.0 

5.9 

5.8 

5.7 

5.6 

5.5 

5.4 

5.4 

5.4 

280 

6.9 

6.8 

6.7 

6.5 

6.3 

6.2 

6.1 

6.0 

5.9 

5.9 

5.9 

5.8 

5.8 

300 

7.1 

7.0 

6.8 

6.8 

6.6 

6.5 

6.4 

6.3 

6.2 

6.2 

6.2 

6.2 

6.2 

320 

7.2 

7.1 

6.9 

6.8 

6.8 

6.7 

6.6 

6.5 

6.5 

6.5 

6.5 

6.6 

6.6 

340 

6.9 

6.9 

7.0 

6.9 

6.9 

6.8 

6.7 

6.8 

6.7 

6.6 

6.7 

6.8 

6.9 

360 

6.9 

6.8 

6.8 

6.8 

6.8 

6.7 

6.7 

6.6 

6.6 

6.8 

6.8 

6.8 

6.9 

380 

6.7 

6.5 

6.5 

6.6 

6.7 

6.6 

6.6 

6.7 

6.7 

6.7 

6.8 

6.9 

6.9 

400 

6.4 

6.4 

6.3 

6.3 

6.4 

6.5 

6.5 

6.5 

6.6 

6.7 

6.7 

6.8 

6.8 

420 

6.0 

6.2 

6.3 

6.3 

6.2 

6.2 

6.3 

6.3 

6.3 

6.3 

6.5 

6.6 

6.7 

440 

5.9 

5.9 

6.0 

6.0 

6.0 

6.0 

6.0 

6.1 

6.0 

6.1 

6.2 

6.2 

6.4 

460 

5.4 

5.5 

5.7 

5.8 

5.8 

5.8 

5.8 

5.8 

5.8 

5.8 

5.9 

6.0 

6.1 

480 

53 

5.3 

5.5 

5.5 

5.5 

5.6 

5.5 

5.6 

5.4 

5.6 

5.7 

5.5 

5.8 

500 

5.0 

5.0 

5.1 

5.2 

5.3 

5.3 

5.3 

5.2 

5.2 

5.2 

5.3 

5.4 

5.4 

520 

4.9 

4.9 

4.9 

4.8 

5.0 

5.1 

5.1 

5.1 

5.1 

5.1 

5.0 

5.0 

5.1 

540 

4.8 

4.8 

4,7 

4.8 

4.8 

4.9 

4.9 

5.0 

4.9 

4.8 

4.8 

4.9 

4.8 

560 

4.7 

4.6 

4.6 

4.7 

4.7 

4.6 

4.7 

4.7 

4.7 

4.7 

4.6 

4.6 

4.6 

580 

4.9 

4.6 

4.5 

4.5 

4.6 

4.5 

4.4 

4.4 

4.5 

4.5 

4.5 

4.4 

4.4 

600 

5.1 

4.9 

4.6 

4.5 

4.4 

4.4 

4.4 

4.3 

4.3 

4.3 

4.3 

4.3 

4.3 

620 

5.3 

5.1 

4.9 

4.7 

4.6 

4.4 

4.3 

4.1 

4.2 

4.2 

4.2 

4.2 

4.1 

640 

5.8 

5.4 

5.2 

5.0 

4.7 

4,6 

4.4 

4.1 

4.1 

4.1 

4.2 

4.2 

4.0 

660 

£.4 

6.0 

5.7 

5.4 

5.0 

4.8 

4.7 

4.5 

1.3 

4.2 

4.2 

4.1 

4.0 

680 

7.1 

6.6 

6.2 

5.7 

5.4 

5.1 

4.9 

4.7 

4.5 

4.4 

4.3 

4.0 

3.9 

700 

7.8 

7.2 

6.8 

6.4 

6.0 

5.6 

5.3 

5.0 

4.7 

4.6 

4.6 

4.3 

4.1 

720 

86 

8.0 

7.6 

7.1 

6.6 

6.2 

5.7 

5.5 

5.2 

4.9 

4.6 

4.6 

4.3 

740 

9.4 

9.0 

3.4 

8.0 

7.4 

6.9 

6.3 

6.0 

5.6 

5.3 

5.0 

4.7 

4.5 

760 

10.3 

9.7 

9.3 

8.6 

8.1 

7.6 

7.2 

6.5 

6.2 

5.8 

5.5 

5.2 

4.9 

780 

10.7 

10.5 

9.9 

9.6 

9.0 

8.5 

7.8 

7.4 

7.0 

6.4 

6.1 

5.7 

5.5 

800 

11.0 

11.0 

10.6 

10.2 

9.9 

9.3 

8.8 

8.1 

7.7 

7.3 

6.7 

6.3 

5.8 

820 

11.3 

11.1 

10.9 

10.6 

10.3 

10.0 

9.6 

9.1 

8.5 

7.9 

7.4 

7.0 

6.6 

840 

11.2 

11.3 

11.2 

11.1 

11.0 

10.7 

10.2 

9.9 

9.4 

8.8 

8.2 

7.7 

7.3 

860 

10.9 

11.1 

11.4 

11.3 

11.3 

11.2 

10.7 

10.4 

9.9 

9.6 

9.2 

8.5 

7.9 

880 

10.4 

10.8 

11.0 

11.3 

11.2 

11.2 

11.2 

10.9 

10.5 

10.3 

9.8 

9.3 

8.7 

900 

9.7 

10.1 

10.6 

11.0 

11.2 

11.2 

11.2 

11.0 

10.9 

10.7 

10.2 

10.0 

9.4 

920 

8.7 

9.3 

9.9 

10.3 

10.8 

11.0 

u.i» 

11.2 

11.2 

11.0 

10.7 

10.4 

10.1 

940 

7.7 

8.2 

8.8 

9.5 

10.1 

10.4 

10.9 

11.0 

11.2 

11.2 

11.0 

10.7 

10.5 

960 

6.5 

7.3 

8.1 

8.6 

9.3 

9.8 

10.2 

10.6 

10.8 

11.1 

11.2 

10.9 

10.8 

990 

5.5 

6.2 

7.0 

7.7 

8.3 

8.9 

9.5 

10.0 

10.4 

10.6 

10.8 

11.0 

10.9 

1000 

4.5 

5.2 

5.9 

6.6 

7.3 

8.0 

8.5 

9.0 

9.5 

100 

10.4 

10.7 

10.9 



440 

450 

460 

470 

480 

490 

500 

510 

620 

530 

540 

550 

560 

TABLE  XXXI. 

Perturbations  produced  by  Mars. 

Arguments  II  and  IV. 

IV. 


35 


II. 

560 

570 

580 

590 

600 

610 

620 

630 

640 

650 

660 

670 

680 

0 

109 

10.8 

10.6 

10.4 

10.3 

10.0 

9.7 

9.2 

8.9 

8.5 

8.1 

7.9 

7.7 

20 

11.4 

10.6 

10.7 

10.6 

10.4 

10.2 

9.9 

9.7 

9.3 

9.0 

8.8 

8.5 

8.1 

40 

9.8 

10.1 

10.4 

10.4 

10.5 

10.3 

10.2 

9.9 

9.6 

9.4 

9.1 

8.9 

8.5 

60 

9.1 

9.4 

9.8 

10.2 

j  10.2 

10.3 

10.2 

10.1 

9.9 

9.6 

93 

9.0 

8.8 

80 

8.2 

8.7 

!  9.0 

9.3 

9.6 

9.8 

10.0 

9.9 

9.8 

9.7 

9.5 

9.3 

91 

100 

7.4 

7.9 

8.4 

8.7 

9.0 

9.4 

9.6 

9.7 

9.8 

9.7 

9.7 

9.5 

9.2 

120 

6.6 

6.9 

7.6 

8.1 

8.3 

8.6 

9.0 

9.2 

9.4 

9.5 

9.5 

|  9.4 

9.3 

140 

6.9 

6.3 

€.8 

7.2 

7.7 

8.0 

8.3 

8.7 

8.9 

9.1 

9.2 

9.3 

9.3 

160 

5.3 

5.8 

6.0 

6.5 

6.9 

7.4 

7.7 

8.0 

8.4 

8.5 

8.8 

8.9 

9.0 

180 

5.0 

52 

5.6 

6.0 

6.3 

6.7 

7.1 

7.2 

7.7 

8.1 

8.1 

8.4 

8.6 

200 

4.8 

5.0 

5.3 

5.4 

5.8 

6.1 

6.5 

6.7 

7.1 

7.3 

7.7 

7.8 

8.0 

220 

5.0 

5.0 

5.1 

5.3 

5.5 

5.7 

6.0 

6.3 

6.6 

6.8 

7.0 

7.3 

7.5 

240 

5.2 

5.2 

5.3 

5.3 

5:4 

5.5 

5.7 

5.9 

6.1 

6.4 

6.6 

6.8 

7.1 

260 

5.4 

5.5 

5.5 

5.5 

5.5 

5.5 

5.5 

5.7 

5.8 

6.0 

6.3 

6.4 

6.5 

280 

5.8 

5.8 

5.8 

5.9 

5.8 

5.8 

5.8 

5.9 

5.9 

5.9 

6.0 

6.1 

6.2 

300 

6.2 

6.1 

6.2 

6.1 

6.1 

6.1 

6.2 

6.1 

6.0 

5.9 

5.9 

6.0 

6.1 

320 

6.6 

6.5 

6.6 

6.6 

6.5 

6.5 

6.6 

6.5 

6.5 

6.3 

6.1 

6.0 

6.0 

340 

6.9 

6.9 

6.9 

7.0 

7.0 

6.9 

6.8 

6.9 

6.9 

6.8 

6.6 

6.5 

6.3 

360 

6.9 

7.0 

7.2 

7.3 

7.3 

7.3 

7.4 

7.3 

7.3 

7.1 

7.1 

7.0 

6.7 

380 

6.9 

7.0 

7.2 

7.4 

7.5 

7.6 

7.7 

7.7 

7.7 

7.6 

7.5 

7.4 

7.2 

400 

6.8 

7.0 

7.1 

7.3 

7.6 

7.9 

8.0 

8.0 

8.1 

8.1 

8.1 

7.9 

7.8 

420 

6.7 

6.9 

7.0 

"7.2 

7.6 

7.8 

8.0 

8.2 

8.3 

8.4 

8.4 

8.5 

8.4 

440 

6.4 

6.6 

6.9 

7.0 

7.3 

7.5 

7.9 

8.2 

8.4 

8.6 

8.8 

8.8 

8.9 

460 

6.1 

6.2 

6.5 

6.9 

7.1 

7.2 

7.6 

8.0 

8.4 

8.7 

9.0 

9.1 

9.2 

480 

5.8 

5.9 

6.0 

6.2 

6.7 

7.1 

7.2 

7.6 

7.9 

8.5 

8.9 

9.2 

9.3 

500 

5.4 

5.5 

5.6 

5.9 

6.1 

6.4 

6.9 

7.2 

7.7 

7.9 

8.4 

9.0 

9.4 

520 

5.1 

5.2 

5.2 

5.3 

5.6 

5.9 

6.3 

6.7 

7.0 

7.6 

8.0 

8.4 

9.0 

540 

4.8 

4.8 

4.8 

5.0 

5.1 

5.4 

5.6 

6.0 

6.4 

6.7 

7.5 

8.1 

8.5 

560 

4.6 

4.5 

4.5 

4.5 

4.7 

.4.8 

5.0 

5.3 

5.8 

6.2 

6.6 

7.1 

7.8 

580 

4.4 

4.3 

4.3 

4.3 

4.3 

4.3 

4.5 

4.7 

5.2 

5.5 

5.9 

6.4 

6.9 

600 

4.3 

4.3 

4.2 

4.1 

4.0 

4.0 

4.1 

4.2 

4.5 

4.8 

5.1 

5.7 

6.2 

620 

4.1 

4.0 

4.0 

3.9 

3.9 

3.8 

3.8 

3.8 

3.8 

4.0 

4.4 

4.9 

5.4 

640 

4.0 

3.9 

4.0 

3.8 

3.8 

3.8 

3.7 

3.5 

3.5 

3.6 

3.8 

4.0 

4.5 

660 

4.0 

4.0 

3.9 

3.8 

3.7 

3.5 

3.5 

3.4 

3.3 

3.3 

3.4 

3.5 

3.7 

680 

3.9 

4.0 

3.9 

3.8 

3.6 

3.5 

3.4 

3.3 

3.2 

3.1 

3.1 

3.1 

3.1 

700 

4.1 

3.9 

3.9 

3.9 

3.7 

3.5 

3.4 

3.3 

3.2 

3.0 

3.0 

3.0 

2.9 

720 

4.3 

4.1 

4.0 

3.9 

3.8 

3.8 

3.5 

3.4 

3.1 

2.9 

2.9 

2.7 

2.7 

740 

4.5 

4.2 

4.2 

4.2 

4.0 

3.7 

3.6 

3.4 

3.3 

3.0 

2.8 

2.6 

2.5 

760 

4.9 

4.7 

4.5 

4.3 

4.2 

4.1 

3.8 

3.6 

3.3 

3.1 

2.9 

2.8 

2.5 

780 

5.5 

5.1 

4.9 

4.5 

4.4 

4.3 

4.1 

3.9 

3.8 

3.4 

3.2 

3.0 

2.7 

800 

5.8 

5.6 

5.2 

5.0 

4.6 

4.5 

4.4 

4.3 

4.1 

3.8 

3.5 

3.1 

2.8 

820 

6.6 

6.1 

5.8 

5.5 

5.3 

5.0 

4.8 

4.6! 

4.4 

4.2 

4.0 

3.6 

3.3 

840 

7.3 

6.8 

6.5 

6.1 

5.7 

5.5 

5.2 

5.0 

4,7 

4.6 

4.3 

4.1 

3.8 

860 

7.9 

7.5 

7.0 

6.7 

6.4 

5.9 

5.8 

5.4 

5.1 

5.0 

4.8 

4.6 

4.4 

880 

8.7 

8.2 

7.8 

7.3 

6.9 

6.6 

6.3 

6.0 

5.7 

5.4 

5.2 

5.0 

4.7 

900 

9.4 

9.0 

8.5 

8.0 

7.6 

7.2 

6.8 

6.6 

6.3 

5.9 

5.6 

5.4 

5.2 

920 

10.1 

9.8 

9.2 

8.7 

8-3 

7.8 

7.4 

7.0 

6.7 

6.4 

6.0 

5.8 

5  7 

940 

10.5 

10.2 

9.8 

9.4 

8.8 

8.5 

8.0 

7.6 

7.3 

6.9 

6.6 

6.2 

61 

960 

10.8 

10.5 

10.2 

10.0 

9.5 

9.1 

8.6 

8.2 

7.8 

7.5 

7.1 

6.8 

66 

980 

10.9 

10.7 

10.3 

10.2 

9.9 

9.6 

9.2 

9.0 

8.5 

8.0 

7.7 

7.4 

72 

rooo 

10.9 

10.8 

10.6 

10.4 

10.3 

10.0 

9.7 

9.2 

8.9 

8.5 

8.1 

7.9 

7.7 

560 

570 

580 

590 

600  | 

610  1 

620 

630 

640 

650 

660 

670 

680 

TABLE  XXXI. 


Perturbations  produced  by  Mars. 

Arguments  II.  and  IV. 

IV. 


II. 

680 

690 

700 

710 

720 

730 

740 

750 

760 

770 

780 

790 

800  ! 

0 

7.7 

7.4 

6.9 

6.8 

6.7 

6.4 

6.1 

5.8 

5.5 

5.2 

4.8 

4.4 

3.7 

20 

8.1 

7.8 

7.4 

7.0 

7.1 

6.9 

6.7 

6.4 

6.1 

5.8 

5.5 

5.1 

4.7 

40 

8.5 

8.3 

7.8 

7.5 

7.2 

7.1 

7.0 

6.9 

6.6 

6.4 

6.1 

5.8 

5.3 

60 

8.8 

8.6 

8.3 

8.1 

7.8 

7.6 

7.5 

7.4 

7.1 

6.9 

6.7 

6.3 

6.0 

80 

9.1 

8.9 

8.7 

8.4 

8.1 

8.0 

7.8 

7.6 

7.4 

7.3 

7.1 

6.9 

6.5 

100 

9.2 

8.9 

8.8 

8.7 

8.6 

8.3 

8.0 

7.7 

7.6 

7.6 

7.6 

7.3 

7.0 

120 

9.3 

9.2 

9.0 

8.7 

8.6 

8.4 

8.2 

8.1 

7.9 

7.8 

7.7 

7.6 

7.5 

;  HO 

9.3 

9.2 

9.0 

9.0 

8.7 

8.5 

8.4 

8.3 

8.0 

7.8 

7.7 

7.7 

7.7 

160 

9.0 

9.0 

8.9 

8.8 

8.7 

8.6 

8.5 

8.4 

8.2 

8.0 

7.9 

7.8 

7-8 

180 

8.6 

8.6 

8.7 

8.7 

8.7 

8.6 

8.5 

8.3 

8.3 

8.0 

8.2 

7.8 

7.9 

200 

8.0 

8.2 

8.3 

8.3 

8.5 

8.4 

8.4 

8.4 

8.2 

8.1 

8.1 

8.1 

7.9 

220 

7.5 

7.7 

7.9 

8.1 

8.2 

8.2 

8.1 

8.2 

8.2 

8.0 

8.1 

8.0 

8.0 

240 

7.1 

7.2 

7.4 

7.5 

7.6 

7.7 

7.8 

7.8 

7.9 

8.0 

8.0 

7.8 

7.8 

260 

6.5 

6.7 

6.9 

7.1 

7.2 

7.3 

7.4 

7.5 

7.6 

7.6 

7.7 

7.7 

7.8 

280 

6.2 

6.3 

6.5 

6.7 

6.7 

6.9 

7.1 

7.2 

7.3 

7.3 

7.3 

7.3 

7.4 

300 

6.1 

6.0 

6.2 

6.4 

6.4 

6.5 

6.6 

6.7 

6.9 

6.9 

6.9 

7.1 

7.1 

320 

6.0 

6.0 

6.0 

6.0 

6.2 

6.1 

6.2 

6.3 

6.5 

6.5 

6.6 

6.6 

6.8 

340 

6.3 

6.2 

6.0 

6.0 

6.0 

6.0 

6.1 

6.1 

6.2 

6.2 

6.3 

6.3 

6.4 

360 

6.7 

6.6 

6.4 

6.1 

6.0 

5.9 

6.0 

5.9 

5.9 

5.9 

6.0 

6.1 

6.2 

380 

7.2 

7.1 

6.8 

6.6 

6.4 

6.2 

6.1 

5.9 

5.8 

5.7 

5.6 

5.8 

5.9 

400 

7.8 

7.7 

7.4 

7.1 

6.8 

6.6 

6.4 

6.1 

6.0 

5.8 

5.6 

5.5 

5.6 

420 

8.4 

8.2 

8.0 

7.8 

7.5 

7.2 

6.8 

6.5 

6.2 

6.0 

5.7 

5.5 

5.4 

440 

8.9 

8.8 

8.7 

8.4 

8.2 

7.8 

7.5 

7.1 

6.6 

6.2 

6.0 

5.7 

5.6 

460 

9.2 

9.2 

9.2 

9.0 

8.8 

8.5 

8.2 

7.9 

7.5 

6.9 

6.5 

6.3 

6.0 

480 

9.3 

9.5 

9.6 

9.6 

9.4 

9.2 

9.1 

8.6 

8.3 

7.8 

7.2 

6.9 

6.5 

500 

9.4 

9.6 

9.8 

10.0 

9.9 

9.8 

9.6 

9.4 

9.1 

8.7 

8.2 

7.6 

7.2 

520 

9.0 

9.5 

9.8 

10.1 

10.2 

10.3 

10.3 

10.0 

9.8 

9.5 

9.1 

8.5 

8.0 

540 

8.5 

9.1 

9.5 

10.0 

10.3 

10.5 

10.6 

10.6 

10.4 

10.1 

9.8 

9.5 

9.0 

560 

7.8 

8.5 

9.0 

9.5 

9.9 

10.4 

10.8 

10.8 

10.9 

10.8 

10.6 

10.2 

9.9 

580 

6.9 

7.6 

8.3 

9.0 

9.7 

10.0 

10.4 

10.7 

11.1 

11.2 

11.0 

11.0 

10.6 

600 

6.2 

6.8 

7.4 

8.0 

8.9 

9.6 

10.1 

10.4 

10.9 

11.3 

11.4 

11.3 

11.2 

620 

5.4 

5.9 

6.5 

7.1 

7.8 

8.6 

9.4 

10.3 

10.6 

11.0 

11.5 

11.7 

11.7 

640 

4.5 

5.0 

5.5 

6.2 

6.8 

7.6 

8.4 

9.2 

10.0 

10.7 

11.1 

11.6 

11.8 

660 

3.7 

4.1 

4.7 

5.2 

5.9 

6.5 

7.3 

8.3 

9.1 

9.8 

10.5 

11.2 

11.5 

680 

3.1 

3.4 

3.8 

4.3 

4.8 

5.5 

6.2 

7.0 

7.8 

8.7 

9.6 

10.2 

11.0 

700 

2.9 

2.8 

3.0 

3.4 

3.9 

4.5 

5.2 

6.0 

6.7 

7.5 

8.5 

9.4 

10.1 

720 

2.7 

2.6 

2.5 

2.7 

3.1 

3.5 

4.0 

4.8 

5.6 

6.4 

7.3 

S.2 

9.1 

740 

2.5 

2.4 

2.4 

2.4 

25 

2.7 

3.1 

3.6 

4.5 

5.2 

6.1 

6.9 

7.8 

760 

2.5 

2.3 

2.2 

2.1 

2.1 

2.3 

2.4 

2.8 

3.2 

4.1 

4.7 

5.7 

6.6 

780 

2.7 

2.5 

23 

2.1 

2.0 

1.9 

2.1 

2.2 

2.5 

2.9 

3.6 

4.4 

5.2 

800 

2.8 

2.7 

2.4 

2.2 

2.0 

1.8 

1.8 

1.8 

2.0 

2.3 

2.5 

3.2 

4.0 

820 

3.3 

3.0 

2.7 

2.3 

2.1 

1.9 

1.8 

1.5 

1.7 

1.7 

2.0 

2.2 

2.9 

840 

3.8 

3.5 

3.0 

2.6 

2.3 

2.1 

1.9 

1.6 

1.5 

1.5 

1.6 

1.7 

2.2 

860 

4.4 

4.0 

3.5 

3.2 

2.8 

2.3 

1.9 

1.7 

1.4 

1.3 

1.2 

1.4 

1.6 

880 

4.7 

4.4 

4.1 

3.7 

3.3 

3.0 

2.5 

2.1 

1.7 

1.4 

1.3 

1.2 

1.2 

900 

5.2 

5.0 

4.6 

4.3 

4.0 

3.6 

3.2 

2.7 

2.2 

1.6 

1.3 

1.2 

1.1 

920 

I  5.7 

5.3 

5.1 

5.0 

4.6 

4.2 

3.8 

*3.4 

2.9 

2.3 

1.9 

1.3 

1.1 

940 

6.1 

5.9 

5.6 

5.4 

5.2 

4.8 

4.5 

3.9 

3.5 

3.1 

2.6 

2.1 

1.5 

960 

6.6 

6.4 

6.2 

5.9 

5.6 

5.4 

5.1 

4.7 

4.3 

3.7 

3.2 

2.8 

2.3 

980 

|  7.2 

6.9 

6.6 

6.4 

6.2 

5.9 

5.6 

5.3 

5.0 

4.6 

4.0 

3.5 

3.0 

1000 

7.7 

7.4 

6.9 

e.a 

6.7 

6.4 

6.1 

5.8 

5.5 

5.2 

4.8 

4.4 

3.7 

680 

690 

700 

710 

720 

730 

740 

750 

760 

770 

780 

790 

800 

TABLE  XXXI. 


Perturbations  produced  by  Mars. 

Arguments  II.  and  IV. 

IV. 


II. 

$00 

810 

820 

830  840  850  i  860 

870 

880  890 

900  910 

920 

//  I  .- 

0 

3.7 

3.2 

2.6 

2.1 

1.7 

'l.3 

0.9 

0.7 

0.7 

1.0 

1.2 

1.6 

22 

20 

4.7 

4.2 

3.6 

3.1 

2.4 

1.9 

1.5 

1.2 

0.8 

0.6 

0.9   1.2 

1  5 

40 

5.3 

4.9 

4.5 

3.8 

3.3 

2.7 

2.0 

1.7 

1.4 

1.0 

0.8  0.9 

1.0 

60 

6.0 

5.7 

5.2 

4.7 

4.1 

3.6 

3.1 

2.6 

2.0 

1.5 

1.2  0.9 

1.0 

80 

6.5 

6.3 

6.0 

5.5 

5.0 

4.6 

4.0 

3.4 

2.7 

2.2 

1.8,  1.5 

1.3 

100 

7.0 

6.7 

6.5 

6.3 

5.9 

5.3 

4,9 

4.4 

3.7 

3.1 

2.5  i  2.1 

1.7 

120 

7.5 

7.3 

7.0 

6.8 

6.5 

6.2 

5.7 

5.1 

4.7 

4.1 

3.5!  2.9 

2.4 

140 

7.7 

7.7 

7.5 

7.3 

7.0 

6.7 

6.4 

6.0 

5.6 

5.1 

4.5l  3.8 

33 

160 

7.8 

7.9 

7.7 

7.6 

7.4 

7.2 

7.0 

6.8 

6.3 

5.8 

5.4  4.8 

4.2 

180 

7.9 

7.8 

7.9 

7.9 

7.7 

7.6 

7.5 

7.1 

7.0 

6.6 

6.1 

5.7 

5.2 

200 

7.9 

7.9 

7.8 

7.9 

7.8 

7.7 

7.6 

7.5 

7.5 

7.1 

6.8 

6.3 

6.1 

220 

8.0 

7.9 

7.8 

7.8 

7.8 

7.8 

7.8 

7.8 

7.6 

7.5 

7.4 

7.1 

6.7 

240  !  7.8 

7.7 

7.7 

7.7 

7.7 

7.7 

7.8 

7.8 

7.7 

7.6 

7.6 

7.5 

7.2 

260 

7.8 

7.7 

7.7 

7.6 

7.7 

7.7 

7.7 

7.7 

7.7 

7.7 

7.8 

.7.8 

7.6 

280 

7.4 

7.4 

7.5 

7.5 

7.5 

7.5 

7.5 

7.5 

7.5 

7.6 

7.6 

7.8 

7.7 

300 

7.1 

7.2 

7.3 

7.3 

7.3 

7.3 

7.3 

7.4 

7.5 

7.4 

7.5 

7.5 

7.7 

320 

68 

6.9 

6.8 

7.0 

7.1 

7.1 

7.1 

7.1 

7.3 

7.3 

7.3 

7.4 

7.4 

340 

64 

6.5 

6.6 

6.6 

6.7 

6.7 

6.8 

6.9 

7.0 

7.1 

7.2 

7.2 

7.2 

360 

6.2 

6.2 

6.2 

6.3 

6.4 

6.4 

6.5 

6.6 

6.7 

6.7 

6.9 

6.9 

7.1 

380 

59 

5.8 

5.8 

5.9 

6.0 

6.1 

6.2 

6.3 

6.4 

6.4 

6.4 

6.6 

6.8 

400 

5.6 

5.6 

5.6 

5.7 

5.7 

5.7 

5.8 

5.9 

5.9 

6.0 

6.1 

6.2 

6.4 

420 

54 

5.4 

5.5 

5.5 

5.5 

5.5 

5.5 

5.5 

5.6 

5.6 

5.6 

5.7 

5.8 

440 

56 

5.3 

5.3 

5.3 

5.3 

5.2 

5.2 

5.2 

5.2 

5.1 

5.0 

5.3 

5.5 

460 

60 

5.6 

5.4 

5.3 

5.2 

5.2 

5.1 

5.0 

5.1 

5.2 

5.2 

5.2 

5.3 

480 

6.5 

6.0 

5.7 

5.4 

5.2 

5.2 

5.1 

4.9 

4.9 

4.9 

4.9 

5.0 

5.0 

500 

7.2 

6.8 

6.3 

5.9 

5.6 

5.3 

5.0 

4.8 

4.9 

4.8 

4.8 

4.8 

4.9 

520 

8.0 

7.4 

7.0 

6.5 

6.1 

5.5 

5.4 

5.1 

4.9 

4.7 

4.7 

4.7 

4.8 

540 

9.0 

8.4 

7.8 

7.3 

6.7 

6.3 

5.8 

5.4 

5.2 

4.9 

4.7 

4.7 

4.7 

560 

9.9 

9.5 

8.8 

8.2 

7.7 

7.1 

6.5 

6.0 

5.7 

5.3 

5.0 

4.8 

4.6 

580 

10.6 

10.2 

9.8 

9.3 

8.8 

8.1 

7.2 

6.8 

6.4 

6.0 

5.6 

5.1 

4,9 

600 

11.2 

11.0 

10.7 

10.3 

9.6 

9.1 

8.5 

7.7 

7.1 

6.4 

6.1 

5.6 

5.3 

620 

11.7 

11.5 

11.4 

11.0 

10.6 

9.9 

9.5 

8.9 

8.1 

7.4 

6.8 

6.3 

5.9 

640 

11.8  11.9 

11.8 

11.7 

11.3 

11.0 

10.4 

9.8 

9.3 

8.5 

7.8 

7.1 

6.6 

660 

n.sUi.e 

12.0 

12.1 

11.9 

11.6 

11.2 

10.8 

10.2 

9.6 

8.9 

8.2 

7.5 

680 

11.0 

11.6 

12.1 

12.2 

12.1 

12.2 

12.1 

11.5 

11.1 

10.6 

10.1 

9.2 

8.5 

700 

10.1 

10.9 

11.6 

12.1 

12.4 

12.3 

12.3 

12.3 

11.9 

11.4 

10.8 

10.4 

9.7 

720 

9.1 

10.0 

10.6 

11.4 

11.9 

12.4 

12.6 

12.5 

12.4 

12.0 

11.6  11.2 

0.8 

740 

7.8 

8.8 

9.7 

10.5 

11.3 

11.8 

12.3 

12.8 

12.6 

12.6 

12.3  11.9 

1.5 

760 

6.6 

7.6 

8.5 

9.4 

10.3 

11.0 

11.7 

12.1 

12.6 

12.8 

12.7!  12.5 

2.1 

780 

5.2 

6.3 

7.1 

8.1 

9.2 

10.1 

10.7 

11.6 

12.0 

12.4 

12.8  12.9 

12.8 

800 

4.0 

4.8 

5.7 

6.7 

7,7 

8.7 

9.7 

10.5 

11.3 

11.9 

12.3  12.5 

12.9 

820 

2.9 

3.6 

4.4 

5.4 

6.4 

7.3 

8.4 

9.5 

10.3 

11.0 

11.7  12.1 

12.5 

840 

2.2 

2.7 

3.3 

4.0 

4.9 

6.0 

7.0 

8.0 

9.1 

10.0 

10.8  11.4 

12.0 

860 

1.6 

1.6 

2.2 

2.9 

3.6 

4.6 

5.6 

6.6 

7.6 

8.6 

9.6 

10.5 

11.2 

880 

1.2 

1.3 

1.5 

1.9 

2.6 

3.3 

4.1 

5.2 

6.1 

7.1 

8.2 

9.2 

10.1 

900 

1.1 

1.1 

1.2 

1.3 

1.7 

2.2 

2.9 

3.8 

4.8 

5.7 

6.8 

7.9 

8.8 

920 

1.1 

1.0 

1.0 

1  i 

111 

1.4 

1.9 

2.6 

34 

4.4 

5.3 

6.3 

7.4 

940 

1.5 

1.1 

0.8 

0.9 

1.0 

1.1 

1.3 

1.6 

2.3 

3.1 

3.9 

5.0 

5.9 

960 

23 

1.7 

1.3 

0.9 

0.7 

0.8 

0.9 

1.2 

1.4 

2.0 

2.8 

3.5 

4.6 

980 

30 

2.5 

1.9 

1.4 

1.2 

1.0 

0.8 

0.9 

1.2 

1.4 

1.7 

2.4 

3.3 

1000 

37 

3.2 

2.6 

2.1 

1.7 

1.3 

0.9 

0.7 

0.7 

1.0 

1.2 

1.6 

2.2 

800 

810 

820 

830 

840 

850 

860 

870 

880 

890 

900 

910 

920 

33 


TABLE  XXXI. 

Perturbations  by  Mars. 
Arguments  II.  and  IV. 
IV. 


TABLE  XXXII. 

Peris,  by  Jupiter 
Arg's.  II.  and  V. 
V. 


II. 

920 

930 

940 

950 

960 

970 

980 

990 

1000 

0 

10 

20  i 

30 

0 

2.2 

3.0 

3.8 

4.8 

5.8 

6.9 

7.8 

8.4 

9.5 

15.3 

15.1 

15.0 

15.0 

20 

1.5 

2.1 

2.6 

3.4 

4.4 

5.5 

6.5 

7.6 

8.7 

14.9 

14.9 

14.7 

14.8 

40 

1.0 

1.4 

1.8 

2.5 

3.2 

4.0 

5.2 

6.0 

7.1 

14.7 

14.6 

14.6 

14.5 

60 

•1.0 

1.1 

1.3 

1.8 

2.3 

3.0 

3.7 

4.8 

5.8 

14.4 

14.4 

14.4 

14.4 

80 

1.3 

1.1 

1.2 

1.4 

1.6 

2.2 

2.7 

3.6 

4.5 

13.4 

13.9 

14.0 

14.2 

100 

1.7 

1.3 

1.2 

1.2 

1.3 

1.6 

2.0 

2.6 

3.3 

13.2 

13.4 

13.6 

13.7 

I  120 

2.4 

2.0 

1.5 

1.4 

1.4 

1.4 

1.7 

1.9 

2.4  : 

12.3 

12.7 

13.0 

13.3 

140 

3.3 

2.8 

2.3 

2.0 

1.7 

1.5 

1.5 

1.8 

2.1 

11.3 

11.8 

12.1 

12.5 

160 

4.2 

3.6 

3.1 

2.6 

2.1 

2.0 

1.7 

1.7 

1.9 

10.2 

10.7 

11.2 

11.7 

180 

5.2 

4.6 

4.0 

3.5 

3^1 

2.5 

2.0 

2.0 

1.9; 

9.1 

9.6 

10.1 

10.7 

200 

G.I 

5.5 

5.0 

4.4 

3.9 

3.5 

2.8 

2.7 

2.9 

7.8 

8.3 

8.9 

9.5 

220 

6.7 

6.3 

5.8 

5.4 

4.9 

44 

3.9 

3.2 

3.0 

6.8 

7.2 

7.7 

8.3 

240 

7.2 

6.9 

6.6 

6.1 

5.6 

53 

4.8 

4.2 

3.7 

5.7 

6.2 

6.6 

7.2 

260 

7.6 

7.5 

7.1 

6.8 

6.5 

6.0 

5.6 

5.2 

4.8 

4.8 

5.2 

5.6 

6.1 

280 

7.7 

7.7 

7.5 

7.3 

7.1 

6.7 

6.3 

5.9 

5.5! 

3.9 

4.1 

4.7 

5.2 

300 

7.7 

7.7 

7.7 

7.7 

7.4 

7.2 

7.0 

6.6 

6.1 

3.4 

3.5 

3.9 

4.3 

320 

7.4 

7.4 

7.6 

7.7 

7.6 

7.6 

7.3 

7.1 

6.9 

3.2 

3.1 

3.4 

3.6 

340 

7.2 

7.2 

7.3 

7.5 

7.7 

7.6 

7.6 

7.6 

7.7 

3.2 

3.0 

3.0 

3.1 

360 

7.1 

7.1 

7.1 

7.2 

7.2 

7.6 

7.6 

7.6 

7.5 

3.5 

3.2 

2.9 

2.9 

380 

6.8 

6.9 

7.0 

7.0 

7.0 

7.1 

7.3 

7.5 

7.5 

4.5 

4.0 

3.4 

3.1 

400 

6.4 

6.6 

6.6 

6.7 

6.7 

6.9 

7.0 

7.1 

7.3 

5.0 

4.3 

3.8 

3.5 

420 

5.8 

5.9 

6.2 

63 

6.6 

6.5 

6.7 

6.7 

6.9 

6.1 

5.2 

4.6 

4.1 

440 

5.5 

5.6 

5.7 

5.8 

6.0 

6.1 

6.3 

6.5 

6.5 

7.5 

6.6 

5.8 

4.9 

460 

5.3 

5.3 

5.4 

5.7 

5.7 

5.7 

5.9 

6.1 

6.2 

9.0 

7.9 

7.0 

6.3 

480 

5.0 

5.0 

5.0 

5.1 

5.3 

5.4 

5.5 

5.6 

5.8 

10.5 

9.5 

8.5 

7.6 

500 

4.9 

4.9 

5.0 

5.0 

5.0 

5.1 

5.2 

5.3 

5.3 

12.3 

11.3 

10.0 

9.1 

520 

4.8 

4.8 

4.8 

4.8 

4.8 

4.7 

4.9 

5.0 

5.l! 

14.0 

12.7 

11.7 

10.7 

540 

4.7 

4.7 

4.6 

4.6 

4.6 

4.5 

4.6 

4.6 

4.7! 

15.6 

145 

13.3 

12.3 

560 

4.6 

4.5 

4.5 

4.4 

4.5 

4.5 

4.5 

4.5 

4.4 

17.1 

16.1 

15.1 

14.0 

580 

4.9 

4.7 

4.6 

4.5 

4.4 

4.4 

4.4 

4.4 

4.2 

18.6 

17.4 

16.5 

15.7 

600 

5.3 

4.9 

4.8 

4.7 

4.5 

4.4 

4.4 

4.3 

4.1 

19.8 

19.0 

17.9 

17.0 

620 

5.9 

5.5 

5.1 

4.8 

4.6 

4.5 

4.4 

43 

4.2 

20.8 

20.1 

19.2 

18.4 

640 

6.6 

6.1 

5.6 

5.4 

5.0 

4.7 

4.6 

4.5 

4.3 

21.6 

20.9 

20.2 

19.5 

660 

7.5 

6.8 

6.3 

5.9 

5.5 

5.3 

4.9 

4.8 

4.6 

22.1 

21.6 

21.0 

20.4 

680 

8.5 

7.8 

7.3 

6.5 

6.1 

5.6 

5.4 

5.1 

4.8 

22.3 

22.0 

21.6 

21.2 

700 

9.7 

8.9 

8.1 

7.6 

7.0 

6.3 

5.9 

5.6 

5.3 

22.2 

22.0 

21.7 

21.5 

720 

10.8 

10.0 

9.3 

8.5 

7.9 

7.2 

6.6 

6.1 

5.8 

22.0 

21.9 

21.7 

21.6 

740 

11.5 

11.0 

10.2 

9.7 

8.9 

8.2 

7.6 

6.9 

6.5 

21.6 

21.6 

21.5 

21.5 

760 

12.1 

11.8 

11.3 

10.5 

10.0 

9.3 

8.5 

7.9 

7.3 

21.2 

21.1 

21.1 

21.2 

780 

12.8 

12.3 

11.9 

11.4 

10.9 

10.2 

9.6 

9.0 

8.2 

20.4 

20.5 

20.6 

20.7 

800 

12.9 

12.9 

12.5 

12.1 

11.7 

11.2 

10.5 

9.8 

9.2 

19.6 

19.8 

19.9 

20.1 

820 

12.5 

12.7 

12.8 

12.7 

12.2 

11.9 

11.2 

10.7 

10.1 

18.8 

19.0 

19.2 

19.4 

840 

12.0 

12.4 

12.6 

12.8 

12.6 

124 

12.2 

11.5 

10.9 

18.1 

18.2 

18.4 

18.6 

860 

11.2 

11.8 

12.3 

12.5 

12.7 

12.5 

12.5 

12.3 

11.7 

17.4 

17.5 

17.6 

17.9 

880 

10.1 

11.0 

11.5 

12.1 

12.3 

12.6 

12.6 

12.4 

12.3 

16.9 

16.9 

16.9 

17.1 

900 

8.8 

9.8 

10.6 

11.3 

11.8 

12.2 

12.4 

12.5 

12.4 

16.3 

16.4 

16.4 

16.5 

920 

7.4 

8.4 

9.3 

10.2 

11.0 

11.5 

12.1 

12.2 

12.3 

16.0 

15.-9 

15.9 

16.0 

940 

5.9 

7.1 

8.1 

8.9 

9.9 

10.7 

11.2 

11.7 

12.1 

15.8 

15.7 

15.7 

15.6| 

960 

4.6 

5.6 

6.7 

7.7 

8.7 

9.4 

10.2 

10.9 

11.4 

15.5 

15.4 

15.3 

15.4  i 

980 

3.3 

4.2 

5.2 

6.2 

7.3 

8.2 

8.9 

9.9 

10.6 

15.3 

15.2 

15.2 

15.1 

1000 

f  2.2 

3.0 

3.8 

4.8 

5.8 

6.9 

7.8 

8.7 

9.5 

15.3 

15.1 

15.0 

15.0 

920 

930 

940 

950 

960 

970 

980 

990 

1000 

0 

10 

20 

30 

TABLE  XXXII. 


39 


Perturbations  produced  by  Jupiter. 

Arguments  II.  and  V. 

V. 


!  n. 

30  1  40 

50 

60  |  70 

80   90 

100  110 

1  120 

130 

140 

150 

0 

15.0 

14.8 

14.7 

14.7 

14.6 

14.5  i  14.5 

14.4 

14.5 

14.5 

14.6 

14.7 

14.8 

20 

14.8 

14.7 

14.6 

14.4 

14.4 

14.2  I  14.2 

14.1  14.1 

14.1 

14.1 

14.1 

14.2 

40 

14.5 

14.4  14.4 

14.3 

14.2 

14.1  13.9 

13.8 

13.8 

13.8 

13.8 

13.8 

13.7 

60 

14.4 

14.3  14.3 

14.2 

14.1 

13.9  13.8 

13.6 

Iflfl 

13.5 

13.5 

13.4!  13.3 

80 

14.2 

14.2  14.1 

14.5 

14.0 

13.8  13.7 

13.5 

13.4 

13.2 

13.1 

13.0!  13.1 

100 

13.7 

13.7  13.9 

13.9 

13.8 

13.7 

13.6 

13.5 

13.4 

13.2 

13.0 

12.8 

12.7 

120 

13.3 

13.4  i  13.4 

13.5 

13.6 

13.5 

13.5 

13.3 

13.3 

13.2 

13.0 

12.8 

12.6 

140 

12.5 

12.8 

13.0 

13.1 

13.2 

13.2 

13.3 

13.2 

13.1  13.0 

12.9 

12.8|  12.6 

160 

11.7 

12.0 

12.4 

12.6 

12.7 

12.8 

12.9 

12.9 

13.0  12.9 

12.8 

12.7 

12.5 

180 

10.7 

11.1 

11.6 

11.9 

12.2 

12.3  12.5 

12.5 

12.6 

12.7 

12.8 

12.6 

12.5 

200 

9.5 

10.0 

10.6 

11.0 

11.5 

11.7 

11.9 

12.2 

12.2 

12.3 

12.4 

12.3 

12.3 

220 

8.3 

8.8 

9.5 

9.9 

10.4 

10.8 

11.3  11.5 

11.8 

11.9 

12.0 

12.0 

12.0 

240 

7.2 

7.7 

8.2 

8.9 

9.4 

9.8 

10.3 

10.6 

11.0 

11.3 

11.5 

11.7 

11.8 

260 

6.1 

6.5 

7.1 

7.6 

8.3 

8.8 

9.3 

9.7 

10.1 

10.5 

10.9 

11.0 

11.2 

280 

5.2 

5.5 

6.0 

6.5 

7.1 

7.6 

8.2 

8.7 

9.2 

9.6 

10.0 

10.4 

10.6 

300 

4.3 

4.7 

5.1 

5.5 

6.1 

6.6 

7.1 

7.6 

8.1 

8.7 

9.1 

9.4 

9.9 

320 

3.6 

3.9 

4.3 

4.6 

5.1 

5.4 

6.0 

6.6 

7.2 

7.7 

8.1 

8.5 

8.9 

340 

3.1 

3.3 

3.5 

3.8 

4.1 

4.5 

5.0 

5.4 

6.1 

6.6 

7.2 

7.6 

8.0 

360 

2.9 

3.0   3.1 

3.3 

3.6 

3.8 

4.1 

4.5 

5.0 

5.5 

6.1 

6.6 

7.1 

380 

3.1 

2.8 

2.8 

2.7 

2.8 

2.9 

3.0 

3.2 

3.5 

4.1 

4.6 

5.0 

5.6 

400 

35 

3.1 

2.9   2.9 

2.8 

2.8 

3.0 

3.1 

3.4 

3.8 

4.2 

4.7 

5.2 

420 

4.1 

3.6  3.3 

3.1 

2.8 

2.7 

2.8 

2.9 

3.1 

3.2 

3.5 

3.8 

4.3 

440 

4.9 

4.4  3.9 

3.4 

3.1 

2.7 

2.8 

2.7 

2.8 

3.1 

3.1 

3.2 

3.5 

460 

6.3 

5.4  4.8   4.3 

3.7 

3.2 

2.9 

2.8 

2.8 

2.7 

2.7 

2.8 

3.2 

480 

7.6 

6.7  i  5.9   5.2 

4.6 

4.1 

3.6 

3.1 

3.0 

2.8 

2.8 

2.6 

2.7 

500 

9.1 

8.1 

7.2   6.4 

5.7 

5.0 

4.4 

3.9 

3.4 

3.2 

3.1 

2.9 

2.7 

520 

10.7 

9.5 

8.7 

7.7 

6.9 

6.1 

5.5 

4.8 

4.2 

3.8 

3.5 

3.2 

3.1 

540 

12.3 

11.1 

10.2 

9.1 

8.4 

7.4 

6.6 

5.9 

5.3 

4.7 

4.1 

3.8 

3.5 

560 

14.0 

13.0 

11.9 

10.8 

9.9 

8.7 

7.9 

7.1 

6.4 

5.8 

5.2 

4.5  1  4.1 

580 

15.7 

14.5 

13.6 

12.5 

11.4 

10.4 

9.3 

8.3 

7.7 

6.9 

6.2 

5.5 

5.0 

600 

17.0 

16.0 

15.0 

14.0 

13.1 

12.0 

11.0 

10.1 

9.2 

8.3 

7.5 

6.7 

6.0 

620 

18.4 

17.4 

16.5 

15.5 

14.7 

13.6 

12.6 

11.6 

10.7 

SS 

9.0 

8.0 

7.3 

640 

19.5 

18.5 

17.9 

17.0 

16.0 

15.1 

14.2 

13.1 

12.2 

IT.J 

10.8 

9.4 

8.7 

660 

20.4 

19.7 

18.9 

18.1 

17.4 

16.3 

15.6 

14.6 

13.7 

12.8 

11.9 

li.O 

10.1 

680 

21.2 

20.5 

19.9 

19.1 

18.5 

17.6 

16.8 

16.0 

15.1 

14,2 

13.5 

12.5 

11.6 

700 

21.5 

21.0 

20.6 

200 

19.3 

18.7 

18.0 

17.1 

16.5 

15.6 

14.7 

13.8 

13.0' 

720 

2L6 

21.2 

21.0 

20.5 

20.0 

19.3 

18.9 

18.3 

17.5 

16.8 

16.1 

15.1 

14.3 

740 

21.5 

21.2 

21.1 

20.8 

20.5 

20.0 

19.4 

18.9 

18.4 

17.7. 

17.2 

16.8 

15.7 

760 

21.2 

21.0 

21.0 

20.8 

20.7 

20.3 

20.0 

19.4 

19.0 

18.6 

17.9 

17.4 

16.7 

7SO 

20.7 

20.7 

20.7 

20.6 

20.6 

20.3 

20.2 

19.8 

19.4 

19.1 

18.7 

18.1 

17.6 

8uO 

20.1 

20.2 

20.3 

20.3 

20.4 

20.3 

20.1 

19.9 

19.7 

19.3 

19.1 

18.7 

18.2 

820 

19.4 

19.5 

19.7 

19.8 

19.9 

19.9 

199 

19.8 

19.8 

-19.6 

19.2 

18.9| 

18.7 

840 

18.6 

18.8 

1S.9 

19.0 

19.2 

19.3 

194 

19.4 

19.4 

19.4 

19.4 

19.0  18.9 

860 

17.9 

18.0 

18.3 

18.4 

18.6 

18.7 

18.8 

18.9 

19.0 

19.1 

19.1 

19.0(18.8 

880 

17.1 

17.2 

17.5 

17.6 

17.9  18.0 

18.2 

18.3 

18.5 

18.6 

18.6 

18.6 

18.7 

900 

16.5 

16.6 

16.8 

16.9 

17.1  17.1 

17.4 

17.5 

17.7 

17.9 

18.1 

18.2 

18.2 

920 

16.0 

16.0 

16.1 

16.2 

16.4  16.5 

16.7 

16.8 

17.0 

17.2 

17.4 

17.5 

17.7 

940 

15.6 

15.5 

15.6 

15.6 

15.7  15.8 

16.0 

16.1 

163 

16.5 

16.8 

16.8 

17.1 

960 

15.4 

15.3 

15.3  15.2 

15.2  15.2 

15.3 

15.4 

15.6 

15.7 

15.9 

16.0 

16.3 

980 

15,1 

15.0 

15.0 

14.9 

14.9  14.8 

14.9 

14.9 

14.9 

15.0 

15.2 

15.3 

15.5 

1000 

15.0 

14.8 

14.7 

14.7 

14.6  14.5 

14.5 

14.4 

14.5 

14.5 

14.6 

14.7 

14.8 

30 

40 

50 

60 

70   80 

90 

100 

110 

120 

130 

140 

150 

TABLE  XXXII. 


Perturbations  produced  by  Jupiter. 
Arguments  II.  and   V. 
V. 


II. 

150 

160 

170 

180  190 

200 

2101  220 

230 

240  |  250 

260 

270 

0 

14.8 

15.0 

15.3 

15.5  15.8 

15.9 

16.2 

16.3 

16.7 

17.0 

17.1 

17.3 

17.5 

20 

14.2 

14.3 

14.6 

H.8  14.9 

15.2  15.5 

15.7 

15.9 

16.2 

16.6 

16.8 

17.1 

40 

13.7 

13.7 

13.9 

14.1  14.3 

14,5  14.8 

15.0 

15.3 

15.5 

15.8 

16.2 

16.4 

60 

13.3 

13.2 

13.4 

13.5  113.6 

13.8  14.1 

14.3 

14.6 

14.8 

15.1 

15.5 

15.8 

80 

13.1 

13.0 

13.0 

13.0 

13.1 

13.1 

13.3 

13.5 

13.8 

14.1 

14.4 

14.5 

15.1 

100 

12.7 

12.7 

12.7 

12.6 

12.7 

12.6 

12.8 

12.9 

13.1 

13.4 

13.7 

14.0 

14.2 

120 

12.6 

12.5 

12.5 

12.4 

12.3 

12.2 

12.3 

12.3 

12.6 

12.8 

13.0 

13.3 

13.6 

140 

12.6 

12.4 

12.4 

12.3 

12.1 

12.0 

12.0 

12.0 

12.1 

12.1 

12.3 

12.5 

12.8 

160 

12.5 

12.3 

12.2 

12.1 

12.1 

11.9 

11.8 

11.8 

11.8 

11.8 

11.9 

12.0 

12.2 

180 

12.5 

12.3 

12.2 

12.1 

11.9 

11.8 

11.7 

11.5 

11.5 

11.5 

11.6 

11.7 

11.8 

200 

12.3 

12.2 

12.2 

12.0 

11.9 

11.7 

11.7 

11.5 

11.4 

11.3 

11.2 

11.3 

11.5 

220 

12.0 

12.0 

12.1 

12.0 

11.8 

11.6 

11.6 

11.5 

11.4 

11.3 

11.2 

11.1 

11.1 

240 

11.8 

11.8 

11.9 

11.9 

11.8 

11.6 

11.5 

11.4 

11.3 

11.2 

11.1 

11.1 

11.0 

260 

11.2 

11.5 

11.6 

11.6 

11.6 

11.5 

11.3 

11.3 

11.3 

11.2 

11.1 

11.0 

10.9 

280 

10.6 

10.8 

11.1 

11.2 

11.2 

11.2 

11.3  11.3 

11.2 

11.2 

11.1 

11.0 

10.9 

300 

9.9 

10.1 

10.5 

10.8 

10.9 

11.0 

11.1 

11.0 

11.0 

11.0 

11.0 

11.1 

10.9 

320 

8.9 

9.4 

9.7 

10.1 

10.4 

10.5 

10.7 

10.8 

10.8 

10.8 

10.8 

10.8 

10.9 

340 

8.0 

8.5 

9.1 

9.3 

9.6 

9.9 

10.2 

10.3 

10.5 

10.6 

10.6 

10.7 

10.7 

360 

7.1 

7.5 

8.0 

8.4 

8.9 

9.2 

9.5 

9.8 

10.1 

10.3 

10.4 

10.5 

10.5 

380 

5.6 

6.2 

6.8 

7.3 

7.8 

8.3 

8.9 

9.3 

9.7 

10.0 

10.0 

10.1 

10.2 

400 

5.2 

5.6 

6.2 

6.6 

7.0 

7.5 

7.9 

8.4 

8.8 

9.1 

9.4 

9.7 

9.9 

420 

4.3 

4.8 

5.3 

5.8 

6.2 

6.6 

7.1 

7.4 

7.9 

8.4 

8.7 

9.1 

9.4 

440 

3.5 

3.9 

4.4 

4.9 

5.4 

5.7 

6.2 

6.7 

7.1 

7.6 

7.9 

8.4 

8.7 

460 

3.2 

3.3 

3.8 

4.1 

4.5 

4.9 

5.4 

5.7 

6.3 

6.7 

7.2 

7.7 

8.0 

480 

2.7 

2.9 

3.2 

3.6 

3.9 

4.3 

4.7 

5.0 

5.4 

5.9 

6.3 

6.8 

7.3 

500 

2.7 

2.7 

2.9 

3.1 

3.4 

3.6 

4.0 

4.4 

4.8 

5.2 

5.7 

5.9 

6.4 

520 

3.1 

2.8 

2.9 

3.0 

3.1 

3.2 

3.5 

3.8 

4.2 

4,7 

4.9 

5.4 

5.7 

540 

3.5 

3.2 

3.1 

3.0 

3.0 

3.0 

3.3 

3.5 

3.7 

4.1 

4.3 

4.7 

5.1 

560 

4.1 

3.8 

3..6 

3.3 

3.2 

3.2  j  3.2 

3.3 

3.5 

3.7 

4.0 

4.3 

4,5 

580 

5.0 

4.6 

4.2 

4.0 

3.6 

3.5  '  3.3 

3.2 

3.4 

35 

3.7 

4.0 

4.2 

600 

6.0 

5.4 

5.1 

4.6 

4.3 

3.9   3.7 

3^5 

3.5 

3.6 

3.7 

3.8 

4.0 

620 

7.3 

6.6 

6.0 

5.6 

5.1 

4.6  ,  4.2 

4.0 

3.9 

3.8 

3.9 

3.9 

4.0 

640 

8.7 

7.8 

7.3 

6.6 

6.1 

5.5 

5.2 

4.7 

4.4 

4.2 

4.0 

4.0 

4.1 

660 

10.1 

9.3 

8.6 

7.7 

7.2 

6.5 

6.2 

5.9 

5.3 

4.9 

4.6 

4.5 

4.4 

680 

11.6 

10.8 

10.0 

9.3 

8.5 

7.5 

7.3 

6.7 

6.3 

5.8 

5.5 

5.2 

4.9 

700 

13.0 

12.1 

11.5 

10.7 

9.9 

9.0 

8.5 

7.8 

7.4 

6.9 

6.3 

6.0 

5.8 

720 

14.3 

13.5 

12.8 

12.1 

11.3 

10.6 

9.8 

9.1 

8.7 

8.0 

7.6 

7.0 

6.6 

740 

15.7 

14.9 

14.2 

13.4 

12.7 

12.0 

11.2 

10.5 

9.7 

9.3 

8.9 

8.2 

7.7 

760 

16.7 

15.9 

15.5 

14.7 

13.9 

13.3 

12.6 

11.8 

11.2 

10.5 

10.0 

9.5 

9.0 

780 

17.6 

17.0 

16.4 

15.7 

15.1 

14.6 

13.8 

13.2  |  12.6 

11.9 

11.2 

10.3 

10.S 

800 

18.2 

17.8 

17.3 

16.8 

16.2 

16.0 

15.0 

14.3 

13.7 

13.1 

12.6 

12.0 

11.5 

820 

18.7 

18.3 

18.0 

17.6 

17.0 

16.6 

16.0 

15.3 

14.9 

14.3 

13.7 

13.1 

12.6 

840 

18.9 

18.7 

18.4 

18.2 

17.7 

17.2 

16.8 

16.3 

15.8 

15.3 

14.9 

14.4 

13.8 

860 

18.8 

18.7 

18.6 

18.4 

18.3 

17.9 

17.4 

17.1 

16.7 

16.3 

15.9 

15.4 

15.0 

8SO 

18.7 

18.5 

18.6 

18.5 

18.3 

18.2 

18.0 

17.7 

17.4 

17.1 

16.6 

16.3 

15.9 

900 

18.2 

18.2 

18.3 

18.3 

18.3 

18.1 

18.1 

18.0 

17.8 

17.6 

17.3 

17.0 

16.7 

1 

920 

17.7 

17.9 

18.0 

18.0 

18.1 

18.1 

18.0 

18.0 

18.0 

17.8 

17.7 

17.6 

17.3 

940 

17.1 

17.1 

17.4 

17.6 

17.6 

17.7 

17.8 

17.8 

17.9 

18.0 

17.8 

17.8 

17.7 

960 

16.3 

16.5 

16.8 

1  j.9 

17.1 

17.2 

17.4 

17.5 

17.6 

17.8 

17.9 

18.0 

17.9 

980 

15.5 

15.7 

16.1 

16.3 

16.5 

16.7 

16.8 

17.0 

17.2 

17.3 

17.6 

17.7 

17.9 

1000 

14.8 

15.0 

15.3 

15.5 

15.8 

15.9 

16.2 

16.3 

16.7 

17.0 

17.1 

17.3 

17.5 

150 

160 

170 

180 

190 

200 

210 

220 

230  240 

250 

260 

270 

TABLE  XXXII. 


41 


Perturbations  produced  by  Jupiter. 
Arguments  II.  and  V 
V 


II. 

270  280  290 

300 

310 

320 

330 

340  350 

360  370 

380 

390 

0 

17.5 

17.5 

17.7 

17.8 

17.9 

17.9 

18.0 

18.0  17.9 

17.7 

17.6 

17.5 

17.5 

20 

17.1 

17.3 

17.5 

17.6 

17.8 

17.8 

18.0 

18.1 

18.1 

18.1 

18.0 

18.0 

18.0 

40 

16.4 

16.8 

16.9 

17.2 

17.6 

17.7 

17.9 

18.1 

18.3 

18.3  18.4 

18.4 

18.6 

60 

15.8 

16.0 

16.4 

16.7 

16.9 

17.3 

17.6 

17.9 

18.2 

18.3 

18.5 

18.5 

18.7 

80 

15.1 

15.4 

15.7 

J6.1 

16.4 

16.7 

17.0 

17.5 

17.8 

18.0 

18.3 

18.5 

18.8 

100 

14.2 

14.6 

15.1 

15.0 

15.8 

16.1 

16.5 

17.0 

17.2 

17.5 

17.9 

18.3 

18.7 

120 

13.6 

13.7 

14.2  !  14.5 

15.0 

15.4 

15.8 

16.2 

16.7 

17.1 

17.3 

17.9 

18.3 

140 

12.8 

13.1 

13.3 

13.7 

14.2 

14.4 

15.1 

15.5 

15.9 

16.3 

16.8 

17.3 

17.7 

160 

12.2 

12.4 

12.6 

12.9 

13.4 

13.8 

14.1 

14.6 

15.2 

15.5 

16.0 

16.5 

17.1 

180 

11.8 

11.9 

12.1 

12.3 

12.5 

12.8 

13.3 

13.7 

14.4 

14.7 

15.2 

15.7 

16.3 

200 

1.1.5 

11.5 

11.6 

11.7 

12.0 

12.1 

12.5 

13.0 

13.4 

13.8 

14.3 

14.7 

15.5 

220 

11.1 

11.1 

11.2 

11.3 

11.6 

11.7 

11.9 

12.3 

12.7 

13.0 

13.5 

14.0 

14.5 

240 

11.0 

10.9  10.9 

11.0 

11.2 

11.3 

11.5 

11.8 

12.1 

12.3 

12.8 

13.2 

13.8 

260 

10.9 

10.8 

10.8 

10.8 

10.9 

10.9 

11.1 

11.3 

11.4 

11.6 

12.0 

12.3 

13.0 

280 

10.9 

10.8 

10.7 

10.6 

10.7 

10.6 

10.8 

11.0 

11.2 

11.3 

11.5 

11.8 

12.2 

300 

10.9 

10.8 

10.7 

10.6 

10.6 

10.5 

10.6 

10.7 

10.8 

10.9 

11.1 

11.4 

11.8 

320 

10.9 

10.7 

10.7 

10.6 

10.6 

10.5 

10.5 

10.6 

10.7 

10.6 

10.7 

11.0 

11.2 

340 

10.7 

10.7 

10.6 

10.5 

10.5 

10.4 

10.5 

10.5 

10.6 

10.5 

10.6 

10.7 

10.8 

360 

10.5 

10.5 

10.5 

10.5 

10.5 

10.4 

10.4 

10.4 

10.4 

10.3 

10.5 

10.6 

10.8 

380 

10.2 

10.3  1  10.3 

10.3 

10.4 

10.3 

10.4 

10.4 

10.4 

10.3 

10.3 

10.4 

10.6 

400 

9.9 

10.0 

10.0 

10.2 

10.3 

10.2 

10.2 

10.3 

10.4 

10.3 

10.3 

10.3 

10.5 

420 

9.4 

9.6 

9.8 

9.9 

10.1 

10.2 

10.1 

10:2 

10.2 

10.2 

10.3 

103 

10.4 

440 

8.7 

9.0 

9.2 

9.4 

9.7 

9.8 

10.0 

10.1 

10.2 

10.1 

10.1 

10.2 

10.4 

460 

8.0 

8.4 

8.6 

8.8 

9.1 

9.3 

9.6 

9.9 

10.1 

10.0 

10.0 

10.2 

10.3 

480 

7.3 

7.6 

7.9 

8.4 

8.7 

8.9 

9.1 

9.4 

9.6 

9.7 

9.8 

10.0 

10.1 

500 

6.4 

6.9 

7.2 

7.6 

8.0 

8.3 

8.G 

8.9 

9.2 

V 

9.5 

9.7 

9.9 

520 

5.7 

6.1 

6.6 

6.9 

7.3 

7.6 

7.9 

8.3 

8.6 

8.9 

9.1 

9.4 

9.7 

540 

5.1 

5.4 

5.8 

6.2 

6.7 

7.0 

7.4 

7.7 

8.0 

8.3 

8.6 

8.9 

9.2 

560 

4.5 

4.9 

5.1 

5.5 

6.0 

6.3 

6.7 

7.2 

7.5 

7.7 

8.0 

8.3 

8.7 

580 

4.2 

4.4 

4.8 

5.0 

5.3 

5.7 

6.1 

6.6 

6.9 

.7.1 

7.4 

7.7 

8.1 

600 

4.0 

4.2 

4.3 

4.7 

4.9 

5.2 

5.6 

6.0 

6.3 

6.5 

6.8 

7.2 

7.6 

620 

4.0 

4.0 

4.1 

4.3 

4,7 

4.8 

5.1 

5.5 

5.8 

6  1 

6.4 

6.7 

7.0 

640 

4.1 

4.1 

4.2 

4.2 

4.4 

4.6 

4.8 

5.1 

5.4 

5.6 

5.9 

6.3 

6.6 

660 

4.4  4.3 

4.3 

4.3 

4.5 

4.5 

4.7 

4.9 

5.1 

5.3 

5.5 

5.8 

6.2 

680 

4.9 

4.9 

4.7 

4.6 

4.7 

4.5 

4.6 

4.8 

5.0 

5.1 

5.3 

5.5 

5.8 

700 

5.8 

5.4 

5.2 

5.1 

5.0 

4.9 

4.9 

4.9 

5.1 

5.2 

5.3 

5.4 

5.6 

720 

6.6 

6.2 

5.9 

5.7 

5.6 

5.5 

5.4 

5.3 

5.3 

5.3 

5.3 

5.4 

5.5 

740 

7.7 

7.2 

6.8 

6.5 

6.4 

6.1 

6.0 

5.9 

5.8 

5.7 

5.6 

5.5 

5.7 

760 

9.0 

8.2 

7.9 

7.5 

7.2 

6.9 

6.7 

6.5 

6.3 

6.1 

5.9 

5.9 

6.0 

780 

10.2 

9.7 

9.1 

8.4 

8.2 

7.7 

7.6 

7.4 

7.2 

6.9 

6.6 

6.5 

6.5 

800 

11.5 

11.0 

10.4 

9.8 

9.4 

8.7 

8.5 

8.3 

8.0 

7.7 

7.6 

7.3 

7.1 

820 

12.6 

12.1 

11.7 

11.2 

10.6 

10.1 

9.7 

9.2 

9.1 

8.6 

8.3 

8.1 

7.9 

840 

13.8 

13.2 

12.8 

12.3 

11.9 

11.3 

10.9 

10.5 

10.2 

9.6 

9.4 

9.1 

8.9 

860 

15.0 

14,4 

13.8 

13.5 

13.1 

12.6 

12.1 

11.7 

11.2 

10.7 

10.4 

10.1 

10.0 

880 

15.9 

15.4 

15.0 

14.4 

14.2 

13.7 

13.4 

129 

12.5 

12.0 

11.5 

113 

11.1 

900 

16.7 

16.4 

15.9 

15.5 

15.2 

14.8 

14.4 

14.1 

13.7 

13.2 

12.8 

12.4 

12.2 

920 

17.3 

17.1 

16.8 

16.5 

16.2 

15.7 

15.5 

15.2 

14.8 

14.3 

14.0 

13.6 

13.3 

940 

17.7 

17.5 

17.3 

17.1 

16.9 

16.6 

16.3 

16.1 

16.0 

15.5 

15.0 

14.7 

14.5 

960 

17.9 

17.8 

17.6 

17.5 

17.4 

17.2 

17.0 

16.9 

1.6.8 

16.4 

16.2 

15.8 

15.6 

980 

17.9 

17.8 

17.8 

17.8 

17.8 

17.8  17.6 

17.5 

17.3 

17.2 

17.0 

16.8 

.6.6 

1000 

17.5 

17.7 

17.7 

17.8 

17.9 

17.9  18.0 

18.0 

17.9 

17.7 

17.6 

17.5 

17.5 

27C  |  280 

290  ,300 

310 

320  330 

340 

350 

360 

370 

380 

390 

42 


TABLE  XXXII. 


Perturbations  produced  by  Jupiter. 

Arguments  II.  and  V. 

V. 


II. 

390 

400 

410 

420 

430 

440 

450 

460 

470 

480 

490 

500 

510 

0 

17.5 

17.1 

17.0 

16.7 

16.5 

16.3 

16.1 

15.8 

15.6 

15.1 

14.6 

14.3 

13.9 

20 

18.0 

18.1 

17.7 

17.5 

17.5 

17.2 

17.1 

16.8 

16.7 

16.3 

16.0 

15.6 

15.3 

40 

18.6 

18.6 

18.5 

18.4 

18.3 

18.1 

18.0 

17.8 

17.6 

17.3 

17.2 

16.8 

16.5 

60 

18.7 

18.9 

18.9 

18.9 

18.9 

18.7 

18.8 

18.6 

18.7 

18.4 

18.1 

17.9 

17.7 

80 

18.8 

18.9 

19.2 

19.3 

19.4 

19.3 

19.3 

19.3 

19.3 

19.2 

19.2 

18.9 

18.8 

100 

18.7 

18.9 

19.1 

19.4 

19.7 

19.8 

19.8 

19.8 

19.8 

19.8 

19.9 

19.7 

19.7 

120 

18.3 

18.6 

18.9 

19.2 

19.5 

19.8 

20.0 

20.1 

20.3 

20.3 

204 

20.4 

20.4 

140 

17.7 

18.2 

18.6 

18.9 

19.2 

19.6 

20.0 

20.3 

20.5 

20.6 

20.7 

20.8 

21.0 

160 

17.1 

17.6 

17.9 

18.5 

19.0 

19.3 

19.8 

20.2 

20.5 

20.6 

20.9 

21.1 

21.2 

180 

16.3 

16.8 

17.3 

17.9 

18.3 

18.8 

19.3 

19.8 

20.3 

20.6 

20.9 

21.1 

21.4 

200 

15.5 

16.0 

16.5 

17.1 

17.7 

18.2 

18.6 

19.1 

19.8 

20.2 

20.7 

21.0 

21.4 

220 

14.5 

15.0 

15.6 

16.1 

16.9 

17.4 

18.0 

18.6 

19.0 

19.7 

20.3 

20.7 

21.1 

240 

13.8 

14.2 

14.7 

15.2 

15.9 

16.5 

17.1 

17.7 

18.4 

18.9 

19.5 

20.1 

20.7 

260 

13.0 

13.4 

13.9 

14.4 

15.0 

15.5 

16.3 

16.9 

17.5 

18.0 

18.6 

19.3 

20.0 

280 

12.2 

12.7 

13.0 

13.5 

14.2 

14.7 

15.3 

15.9 

16.7 

17.2 

17.8 

18.4 

19.1 

300 

11.8 

11.9 

12.4 

12.8 

13.3 

13.8 

14.4 

14.9 

15.7 

16.3 

17.0 

17.6 

18.2 

320 

11.2 

11.5 

11.8 

12.2 

12.7 

13.0 

13.6 

14.1 

14.7 

15.3 

16.0 

16.6 

17.4 

340 

10.8 

11.2 

11.4 

11.6 

12.1 

12.4 

12.9 

13.4 

13.9 

14.4 

15.1 

15.7 

16.4 

360 

10.8 

10.8 

11.0 

11.2 

11.6 

11.9 

12.3 

12.6 

13.2 

13.6 

14.2 

14.8 

15.5 

380 

10.6 

10.6 

10.7 

10.9 

11.2 

11.4 

11.9 

12.2 

12.6 

12.9 

13.5 

13.9 

14.5 

400 

10.5 

10.5 

10.6 

10.6 

10.9 

11.1 

11.4 

11.8 

12.2 

12.5 

12.9 

13.3 

13.8 

420 

10.4 

10.4 

10.5 

10.6 

10.7 

10.9 

11.2 

11.3 

11.7 

11.9 

12.4 

12.8 

13.3 

440 

10.4 

10.4 

10.4 

10.5 

10.7 

10.8 

10.9 

11.1 

11.3 

11.6 

11.9 

12.2 

12.7 

460 

10.3 

10.4 

10.4 

10.4 

10.6 

10.6 

10.7 

10.9 

11.2 

11.3 

11.7 

11.9 

12.2 

480 

10.1 

10.2 

10.3 

10.4 

10.6 

10.6 

10.7 

10.8 

11.0 

11.2 

11.4 

11.7 

12.0 

500 

9.9 

10.0 

IP-1 

10.2 

10.4 

10.5 

10.7 

10.8 

10.9 

11.0 

11.2 

11.3 

11.7 

520 

9.7 

9.8 

9.8 

10.0 

10.2 

10.3 

10.5 

10.6 

10.9 

10.8 

11.1 

11.3 

11.5 

540 

9.2 

9.4 

9.6 

9.8 

10.0 

10.2 

10.3 

10.4 

10.6 

10.7 

10.9 

11.1 

11.4 

560 

8.7 

8.9 

9.1 

9.3 

9.7 

9.8 

10.1 

10.3 

10.5 

10.6 

10.7 

10.8 

11.2 

580 

8.1 

8.5 

8.7 

8.7 

9.2 

9.4 

9.7 

9.9 

10.2 

10.4 

10.6 

10.7 

10.9 

600 

7.6 

7.9 

8.2 

8.5 

8.8 

9.0 

9.3 

9.5 

9.8 

10.0 

10.3 

10.5 

10.7 

620 

7.0 

7.3 

7.6 

7.9 

8.2 

8.5 

8.8 

9.0 

9.4 

9.6 

10.0 

10.1 

10.4 

640 

6.6 

6.8 

7.1 

7.4 

7.7 

7.9 

8.2 

8.6 

8.9 

9.1 

9.4 

9.7 

10.1 

660 

6.2 

6.4 

6.6 

6.9 

7.3 

7.6 

7.9 

8.1 

83 

8.6 

8.9 

9.2 

9.5 

680 

5.8 

6.1 

6.2 

6.5 

6.8 

7.0 

7.4 

7.6 

7.9 

8.1 

8.4 

8.7 

9.0 

700 

5.6 

5.8 

6.0 

6.2 

6.4 

6.6 

6.9 

7.1 

7.4 

7.6 

7.9 

8.2 

8.5 

720 

5.5 

5.6 

5.7 

5.9 

6.2 

6.3 

6.5 

6.8 

7.1 

7.2 

7.5 

7.7 

8.0 

740 

5.7 

5.7 

5.7 

5.8 

6.0 

6.1 

6.2 

6.4 

6.7 

6.9 

7.1 

7.2 

7.5 

760 

6.0 

6.0 

6.0 

6.0 

6.0 

6.1 

6.2 

6.3 

6.4 

6.5 

6.7 

6.8 

7.1 

780 

6.5 

6.3 

6.2 

6.2 

6.3 

6.3 

6.3 

6.3 

6.4 

6.4 

6.5 

6.7 

6.8 

800 

7.1 

7.0 

6.7 

6.6 

6.7 

6.5 

6.5 

6.4 

6.5 

6.5 

6.5 

6.6 

«.7 

820 

7.9 

7.6 

7.5 

7.3 

7.2 

7.0 

7.0 

6.8 

6.8 

6.7 

6.6 

6.6 

6.7 

840 

8.9 

8.6 

8.3 

8.1 

7.8 

7.7 

7.6 

7.4 

7.3 

7.1 

7.0 

6.8 

6.8 

860 

10.0 

9.7 

9.3 

9.0 

8.7 

8.4 

8.2 

8.1 

7.9 

7.7 

7.6 

7.3 

7.2 

880 

11.1 

10.5 

10.4 

10.0 

9.7 

9.5 

9.2 

8.9 

8.7 

8.4 

8.2 

7.9 

7.7 

900 

12.2 

11.8 

11.5 

11.0 

10.8 

10.5 

10.3 

9.9 

9.7 

9.4 

9.0 

8.8 

8.5 

920 

13.3 

13.0 

12.6 

12.3 

12.1 

11.5 

11.3 

11.0 

10.6 

10.2 

10.1 

9.7 

9.4 

940 

14.5 

14.1 

13.8 

13.5 

13.2 

12.8 

12.5 

11.9 

11.8 

11.3 

11.0 

10.7 

10.4 

960 

15.6 

15.3 

14.9 

14.6 

14.4 

14.0 

13.7 

13.3 

13.0 

12.5 

12.1 

11.8 

11.5 

980 

16.6 

16.3 

16.0 

15.7 

15.6 

15.2 

14.9 

14.6 

14.2 

13.8 

13.6 

12.9 

12.7 

1000 

17.5 

17.1 

17.0 

16.7 

16.5 

16.3 

16.1 

15.8 

15.6 

15.1 

14.6 

14.3 

13.9 

390 

400 

410 

420 

430 

440 

450 

460 

470 

480 

490 

500 

510 

TABLE  XXXII. 


43 


Perturbations  produced  by  Jupiter. 

Arguments  II.  and  V. 

V. 


II. 

510 

|  520 

530 

540 

550 

|  560 

570 

580 

590 

600 

610 

620 

630 

0 

13.9 

13.4 

13.1 

12.7 

12.1 

11.8 

11.3 

10.8 

10.2 

9.9 

9.4 

I  8.9 

8,4 

20 

15.3 

14.9 

14.4 

13.9 

13.5 

13.1 

12.5 

12.1 

11.5 

11.0 

10.4 

|io.o 

9.4 

40 

16.5 

16.3 

15.7 

15.4 

15.0 

14.3 

13.8 

13.4 

12.8 

12.3 

11.7 

11.1 

10.5 

60 

17.7 

17.3 

17.0 

16.6 

16.1 

15.8 

15.3 

14.7 

14.3 

13.7 

13.0 

12.4 

11.8 

80 

18.8 

18.5 

18.1 

17.9 

17.4 

17.1 

16.6 

16.2 

15.7 

15.1 

14.5 

13.9 

13.2 

100 

19.7 

19.5 

19.2 

19.0 

18.8 

18.4 

17.9 

17.6 

17.0 

16.5 

16.0 

15.2 

14.7 

120 

20.4 

20.3 

20.2 

20.0 

19.7 

19.5 

!19.1 

18.8 

18.4 

18.0 

17.3 

16.8 

16.2 

140 

21.0 

21.1 

21.0 

20.8 

20.7 

20.4 

20.2 

19.9 

19.6 

19.3 

18.8 

18.3 

17.7 

160 

21.2 

21.5 

21.5 

21.6 

21.5 

21.3 

121.2 

21.0 

20.6 

20.4 

20.1 

19.6 

19.1 

180 

21.4 

21.6 

21.8 

22.0 

22.0 

22.1 

21.9 

21.8 

21.6 

21.4 

21.1 

20.7 

20.3 

200 

21.4 

21.7 

21.9 

22.1 

22.3 

22.5 

22.5 

22.5 

22.4 

22.3 

22.1 

21.8 

21.5 

220 

21.1 

21.5 

21.8 

22.2 

22.5 

22.8 

23.1 

23.1 

22.9 

22.8 

22.9 

22.6 

22.5J 

240 

20.7 

21.1 

21.5 

21.  '9 

22.3 

22.7 

23.0 

23.3 

23.4 

23.5 

23.4 

23.3 

23.2J 

260 

20.0 

20.6 

21.0 

21.6 

22.0 

22.4 

22.8 

23.2 

23.5 

23.8 

23.8 

23.8 

23.9 

280 

19.1 

19.9 

20.4 

20.9 

21.5 

22.0 

22.4 

23.0 

23.3 

23.7 

24.0 

24.1 

24.1 

300 

18.2 

19.0 

19.6 

20.3 

20.7 

21.3 

21.8 

22.3 

23.0 

23.4 

23.8 

24.1 

24.3 

320 

17.4 

18.9 

18.7 

19.4 

20.0 

20.6 

21.1 

21.8 

22.3 

22.9 

23.3 

23.7 

24.2 

340 

16.4 

17.0 

17.6 

18.5 

19.2 

19.9 

20.4 

21.1 

21.6 

22.2 

22.8 

23.3 

23.7 

360 

15.5 

16.2 

16.7 

17.4 

18.2 

18.9 

19.5 

20.1 

20.8 

21.5 

22.0 

22.6 

23.2  ; 

380 

14.5 

15.2 

15.9 

16.6 

17.1 

17.9 

18.6 

19.3 

19.8 

20.5 

21.1 

21.8 

22.5 

400 

13.8 

14.4 

14.9 

15.6 

16.2 

16.8 

17.6 

18.4 

19.1 

19.7 

20.3 

20.9 

21.5 

420 

13.3 

13.7 

14.2 

14.8 

15.3 

16.0 

16.5 

17.4 

18.0 

18.7 

19.4 

20.0 

20.6 

440 

12.7 

13.1 

13.6 

14.1 

14.6 

15.2 

15.7 

16.4 

17.1 

17.8 

18.4 

18.9 

19.6 

460 

12.2 

12.7 

13.0 

13.5 

13.9 

14.4 

15.0 

15.6 

16.1 

16.9 

17.5 

18.2 

18.7 

480 

12.0 

12.2 

12.5 

13.0 

13.4 

13.9 

14.3 

14.8 

15.3 

15.9 

16.6 

17.3 

17.9 

500  | 

11.7 

12.0 

12.2 

12.6 

12.9 

13.3 

13.8 

14.3 

14.7 

15.2 

15.7 

16.4 

16.9 

520 

11.5 

11.9 

12.0 

12.3 

12.6 

13.0 

13.2 

13.8 

14.2 

14.7 

15.1 

15.5 

16.2 

540 

11.4 

11.6 

11.9 

12.2 

12.4 

12.7 

12.9 

13.3 

13.7 

14.2 

14.6 

15.0 

15.4 

560 

11.2 

11.4 

11.5 

11.9 

12.1 

12.4 

12.7 

13.1 

13.4 

13.8 

14.1 

14.5 

14.9 

580 

10.9 

11.2 

11.4 

11.6 

11.9 

12.2 

12.4 

12.8 

13.1 

13.5 

13.8 

14.2 

14.5 

600 

10.7 

10.8 

11.1 

11.5 

11.7 

12.0 

12.2 

12.5 

12.8 

13.1 

13.4 

13.8 

14.2 

620 

10.4 

10.7 

10.7 

11.1 

11.4 

11.6 

12.0 

12.3 

12.5 

12.9 

13.1 

13.4 

13.8 

640 

10.1 

10.4 

10.6 

10.7 

11.0 

11.3 

11.6 

12.0 

12.3 

12.6 

12.9 

13.2 

13.5 

660  > 

9.5 

9.9 

10.2 

10.5 

10.6 

11.0 

11.3 

11.6 

11.9 

12.3 

12.6 

12.9 

13.2 

680 

9.0 

9.3 

9.6 

10.0 

10.3 

10.5 

10,8 

11.3 

11.5 

11.9 

12.2 

12.4 

12.8 

700 

8.5 

8.9 

9.1 

9.5 

9.8 

10.1 

10.3 

10.7 

11.1 

11.4 

11.8 

12.1 

12.4 

720 

8.0 

8.3 

8.5 

9.0 

9.2 

9.6 

9.9 

10.2 

10.5 

10.9 

11.3 

11.7 

12.0 

740 

7.5 

7.8 

8.0 

8.3 

8.6 

9.0 

9.3 

9.7 

9.9 

10.4 

10.8 

11.1 

11.5 

760 

7.1 

7.3 

7.5 

7.9 

8.1 

8.4 

8.6 

9.1 

9.4 

9.7 

10.1 

10.5 

10.9 

780 

6.8 

7.0 

7.1 

7.3 

7.6 

7.9 

8.1 

8.5 

8.8 

9.2 

9.4 

9.8 

10.2 

800 

6.7 

6.8 

6.8 

7.0 

7.1 

7.3 

7.5 

7.8 

8.2 

8.5 

8.8 

9.1 

9.5 

820 

6.7 

6.8 

6.6 

6.8 

6.9 

7.0 

7.1 

7.4 

7.6 

7.9 

8.1 

8.4 

8.7 

840 

6.8 

G.8 

6.8 

6.8 

6.8 

6.9 

6.9 

7.1 

7.2 

7.4 

7.6 

7.9 

8.1 

SCO 

7.2 

7.1 

7.1 

7.0 

6.9 

6.9 

6.8 

6.8 

6.9 

7.1 

7.2 

7.3 

7.6 

880 

7.7 

7.5 

7.4 

7.3 

7.1 

7.0 

6.8 

6.8 

6.7 

6.8 

6.8 

7.0 

7.2 

900 

8.5 

8.2 

7.9 

7.7 

7.5 

7.3 

7.2 

7.1 

6.9 

6.9 

6.8 

6.8 

6.8 

920 

9.4 

9.2 

8.7 

8.4 

8.1 

7.9 

7.6 

7.4 

7.1 

7.0 

6.9 

6.8 

6.7 

940 

10.4 

10.0 

9.7 

9.4 

8.9 

8.6 

8.3 

8.1 

7.7 

7.4 

7.1 

6.9 

6.7 

960 

11.5 

11.2 

10.7 

10.4 

9.8 

9.5 

9.1 

8.8 

8.5 

8.1 

7.7 

7.4 

7.1 

980 

12.7 

12.3 

11.8 

11.5 

11.1 

10.6 

10.0 

9.7 

9.2 

8.9 

8.5 

8.1 

7.7 

1000 

13.9 

13.4 

13.1 

12.7 

12.1 

11.8 

11.3 

10.8 

10.2 

9.9 

9.4 

8.9 

8.4 

510 

520 

530 

540 

550 

560 

570 

580 

590 

600 

610 

020 

630  1 

TABLE  XXXII. 


Perturbations  produced  by  Jupiter. 

Arguments  II.  and  V. 

V. 


II. 

630 

640 

650 

660 

670 

680 

690 

700 

710 

720 

730 

740 

750 

0 

8.4 

8.0 

7.7 

7.3 

6.9 

6.7 

6.5 

6.5 

6.3 

6.2 

6.2 

6.4 

6.5 

20 

9.4 

9.0 

8.4 

8.0 

7.5 

7.1 

6.9 

6.7 

6.4 

6.3 

6.0 

6.1 

6.1 

40 

10.5 

10.1 

9.4 

8.9 

8.3 

7.8 

7.4 

7.0 

6.6 

6.4 

6.2 

5.9 

5.8 

60 

11.8 

11.3 

10.6 

10.1 

9.3 

8.7 

8.2 

7.7 

7.2 

6.8 

6.4 

6.2 

5.8 

80 

13.2 

12.7 

12.0 

11.3 

10.5 

9.9 

9.2 

8.7 

8.1 

7.6 

7.1 

6.6 

6.2 

100 

14.7 

14.1 

13.4 

12.8 

12.0 

11.3 

10.6 

9.9 

9.1 

8.5 

7.9 

7.3 

6.8 

120 

16.2 

15.4 

14.9 

14.2 

13.4 

12.7 

12.0 

11.3 

10.4 

9.8 

8.9 

8.2 

7.6 

140 

17.7 

17.2 

16.4 

15.6 

14.9 

14.2 

13.4 

12.7 

11.9 

11.1 

10.2 

9.6 

8.8 

160 

19.1 

18.6 

17.9 

17.3 

16.6 

15.7 

15.0 

14.2 

13.3 

12.6 

11.7 

10.9 

10.0 

180 

20.3 

19.9 

19.4 

18.8 

18.0 

17.3 

16.7 

15.8 

15.0 

14.1 

13.2 

12.4 

11.5 

200 

21.5 

21.2 

20.8 

20.2 

19.3 

18.9 

18.1 

17.5 

16.6 

15.7 

14.9 

14.0 

13.1 

220 

22.5 

22.3 

21.9 

21.5 

21.0 

20.3 

19.7 

19.0 

18.2 

17.5 

16.6 

15.5 

14.7 

240 

232 

23.0 

22.9 

22.5 

22.0 

21.6 

21.1 

20.5 

19.8 

19.1 

18.2 

17.3 

16.4 

260 

23.9 

23.8 

23.7 

23.5 

23.1 

22.7 

22.3 

21.8 

21.2 

20.6 

19.8 

19.1 

18.1 

280 

24.1 

24.3 

24.2 

24.2 

24.0 

23.7 

23.5 

23.1 

22.4 

21.8 

21.2 

20.5 

19.8 

300 

24.3 

24.5 

24.6 

24.6 

24.5 

24.4 

24.2 

23.9 

23.6 

23.1 

22.5 

21.9 

21.2 

320 

24.2 

24.5 

24.7 

24.9 

24.8 

24.8 

24.8 

24.7 

24.4 

24.1 

23.7 

23.1 

22.5 

340 

23.7 

24.2 

24.5 

24.7 

25.0 

25.2 

25.1 

25.0 

25.0 

24.9 

24.6 

24.1 

23.7 

360 

23.2 

23.7 

24.2 

24.5 

24.7 

25.0 

25.1 

25.3 

25.4 

25.3 

25.1 

24.9 

24.5 

380 

22.5 

23.1 

23.6 

24.1 

24.4 

24.7 

25.1 

25.2 

25.4 

25.5 

25.4 

25.3 

25.2 

400 

21.5 

22.3 

22.8 

23.4 

23.9 

24.3 

24.7 

25.1 

25.2 

25.4 

25.6 

25.6 

25.5 

420 

20.6 

21.3 

22.0 

22.6 

23.1 

23.6 

24.1 

24.5 

25.0 

25.2 

25.4 

25.6 

25.7 

440 

19.6 

20.3 

21.0 

21.8 

22.3 

22.9 

23.4 

23.9 

24.3 

24.8 

25.0 

25.2 

25.6 

460 

18.7 

19.4 

20.1 

20.7 

21.3 

21.9 

22.6 

23.3 

23.6 

24.1 

24.6 

24.8 

25.1 

480 

17.9 

18.5 

19.1 

19.7 

20.3 

21.0 

21.6 

22.2 

22.8 

23.3 

23.8 

24.3 

24.6 

500 

16.9 

17.6 

18.2 

18.8 

19.3 

19.9 

20.7 

21.4 

21.9 

22.5 

22.9 

23.4 

23.9 

520 

16.2 

16.8 

17.3 

17.9 

18.4 

19.0 

19.7 

20.4 

21.0 

21.6 

21.1 

22.6 

23.0 

540 

15.4 

16.1 

16.6 

17.2 

17.5 

18.1 

18.7 

19.3 

19.9 

20.5 

21.2 

22.7 

22.2 

560 

14.9 

15.4 

16.0 

16.5 

16.9 

17.3 

17.9 

18.4 

18.9 

19.6 

20.1 

20.7 

21.3 

580 

14.5 

15.0 

15.3 

15.9 

16.3 

16.7 

17.1 

17.6 

18.1 

18.7 

19.3 

19.8 

20.3 

600 

14.2 

14.6 

14.9 

15.3 

15.8 

16.3 

16.6 

17.0 

17.4 

17.9 

18.3 

18.9 

19.4 

620 

13.8 

14.2 

14.6 

14.9 

15.1 

15.7 

16.2 

16.6 

16.9 

17.3 

17.6 

18.0 

18.5 

640 

13.5 

14.0 

14.2 

14.6 

14.8 

15.1 

15.6 

16.1 

16.5 

16.8 

17.1 

17.5 

17.9 

660 

13.2 

13.5 

13.9 

14.3 

14.6 

14.9 

15.2 

15.6 

15.9 

16.4 

16.6 

17.0 

17.3 

680 

12.8 

13.2 

13.5 

13.9 

14.2 

14.5 

14.9 

15.2 

15.6 

16.0 

16.2 

16.5 

16.8 

700 

12.4 

12.9 

13.3 

13.5 

13.8 

14.2 

14.5 

14.9 

15.1 

15.6 

15.9 

16.2 

16.4 

720 

12.0 

12.4 

12.8 

13.2 

135 

13.8 

14.2 

14.5 

14.8 

15.1 

15.5 

15.8 

16.1 

740 

11.5 

11.9 

12.2 

12.6 

12.9 

13.3 

13.8 

14.2 

14.5 

14.8 

15.1 

15.4 

15.7 

760 

10.9 

11.4 

11.8 

12.2 

12.4 

12.8 

13.2 

13.7 

14.1 

14.5 

14.7 

15.0 

15.4 

780 

10.2 

10.6 

11.2 

11.6 

11.9 

12.4 

12.8 

13.2 

13.5 

13.9 

14.3 

14.6 

14.9 

800 

9.5 

10.0 

10.3 

10.9 

11.3 

11.6 

12.1 

12.6 

12.9 

13.4 

13.8 

14.2 

14.5 

820 

8.7 

9.3 

9.7 

10.0 

10.5 

10.9 

11.4 

11.9 

12.3 

12.8 

13.2 

13.6 

14.0 

840 

8.1 

8.4 

8.8 

.9.3 

9.6 

10.1 

10.6 

11.1 

11.6 

12.1 

12.5 

13.0 

13.4 

860 

7.6 

7.9 

8.1 

8.5 

8.8 

9.2 

9.7 

10.2 

10.7 

11.2 

11.7 

12.1 

12.6 

880 

7.2 

7.4 

7.6 

7.8 

8.1 

8.5 

8.8 

9.4 

9.8 

10.2 

10.7 

11.2 

11.8 

900 

6.8 

7.0 

7.1 

7.3 

7.4 

7.8 

8.2 

8.5 

8.9 

9.4 

9.8 

10.3 

10.8 

920 

6.7 

6.8 

6.8 

6.9 

7.0 

7.0 

7.4 

7.8 

8.1 

8.6 

8.9 

9.4 

9.9 

940 

6.7 

6.7 

6.7 

6.8 

6.7 

6.8 

6.8 

7.1 

7.4 

7.7 

8.1 

8.4 

8.9 

960 

7.1 

7.0 

6.8 

17 

6.5 

6.5 

6.6 

6.7 

6.8 

7.1 

7.3 

7.7 

8.0 

980 

7.7 

7.4 

7.1 

6.9 

6.6 

6.5 

6.4 

6.4 

6.3 

6.5 

6.8 

6.9 

7.3 

1000 

8.4 

8.0 

7.7 

7.3 

6.9 

6.7 

6.5 

6.5 

6.3 

6.2 

6.2 

6.4 

6.5 

63) 

140 

650 

660 

670 

680 

690 

700 

710 

720 

730 

740 

750 

TABLE  XXXII. 


45 


Perturbations  produced  by  Jupiter. 

Arguments  II.  and  V. 

V. 


II. 

750 

760 

;  770 

780 

790 

800 

810 

820 

830 

840 

850 

860 

870 

0' 

6, 

6.8 

7.2 

7.5 

8.0 

8.4 

8.8 

9.5 

10.1 

10.5 

11.0 

11.6 

12.4 

20 

6.1 

6.2 

6.5 

6.7 

7.0 

7.4 

7.9 

8.4 

9.0 

9.5 

10.0 

10.6 

11.1 

40 

5.8 

5.9 

5.9 

6.2 

6.4 

6.6 

6.9 

7.4 

7.8 

8.2 

8.8 

9.5 

10.0 

60 

5.8 

5.7 

5.7 

5.7 

5.9 

6.1 

6.2 

6.5 

6.9 

7.2 

7.7 

8.3 

8.8 

80 

6.2 

5.8 

5.7 

5.6 

5.4 

5.6 

5.7 

5.9 

6.1 

6.3 

6.7 

7.3 

7.8 

100 

6.8 

6.3 

5.9 

5.6 

5.5 

5.3 

5.3 

5.4 

5.4 

5.6 

5.9 

6.3 

6.8 

120 

7.6 

7.4 

6.5 

6.0 

5.7 

5.5 

5.1 

5.2 

5.1 

5.1 

5.2 

5.5 

5.8 

140 

8.8 

8.1 

7.4 

6.8 

6.2 

5.8 

5.4 

5.2 

5.0 

4.9 

4.8 

5.0 

5.1 

160 

10.0 

9.3 

8.5 

7.8 

7.2 

6.5 

5.9 

5.5 

5.1 

5.9 

4.7 

4.7 

4.7 

180 

11.5 

10.6 

9.7 

9.0 

8.2 

7.5 

6.9 

6.3 

5.8 

5.2 

4.8 

4.7 

4.5 

200 

13.1 

12.2 

11.2 

10.4 

9.5 

8.8 

7.9 

7.1 

6.5 

5.9 

5.3 

5.0 

4.7 

220 

14.7 

13.8 

12.9 

12.0 

11.1 

10.2 

9.3 

8.4 

7.5 

6.7 

6.1 

5.5 

5.2 

240 

16.4 

15.3 

14.5 

13.6 

12.6 

11.7 

10.7 

9.8 

8.8 

7.9 

7.0 

6.5 

5.9 

260 

18.1 

17.2 

16.3 

15.3 

14.3 

13.3 

12.2 

11.4 

10.4 

9.4 

8.3 

7.7 

6.9 

280 

19.8 

18.9 

17.9 

17.0 

16.1 

15.0 

14.0 

13.0 

11.9 

10.9 

9.9 

8.9 

8.0 

300 

21.2 

20.4 

19.6 

18.7 

17.7 

16.8 

15.8 

14.7 

13.7 

12.6 

11.5 

10.5 

9.4 

320 

22.5 

21.9 

21.2 

20.4 

19.4 

18.5 

17.4 

16.5 

15.5 

14.2 

13.2 

12.3 

11.2 

340 

23.7 

23.0 

22.4 

21.8 

21.1 

20.2 

19.2 

183 

17.1 

16.1 

15.0 

13.9 

12.9 

360 

24.5 

24.0 

23.6 

23.0 

22.4 

21.6 

20.8 

19.9 

18.9 

17.9 

16.8 

15.9 

14.7 

380 

25.2 

24.9 

24.5 

24.0 

23.5 

22.8 

22.1 

21.4 

20.5 

19.5 

18.5 

17.6 

16.5 

400 

25.5 

25.4 

25.1 

24.8 

24.5 

23.9 

23.4 

22.7 

21.9 

21.0 

20.1 

19.2 

18.2 

420 

25.7 

25.6 

25.5 

25.3 

25.0 

24.5 

24.2 

23.7 

23.2 

22.3 

21.5 

20.7 

198 

440 

25.6 

25.6 

25.7 

25.7 

25.5 

25.3 

24.9 

24.6 

24.1 

23.4 

22.7 

22.0 

21.2 

460 

25.1 

25.3 

25.5 

25.6 

25.8 

25.7 

25.4 

25.2 

24.8 

24.3 

23.7 

23.1 

22.5 

480 

24.6 

24.9 

25.2 

25.4 

25.6 

25.6 

25.5 

25.4 

25.2 

24.9 

24.5 

24.1 

23.5 

500 

23.9 

24.2 

24.7 

25.0 

25.3 

25.4 

25.5 

25.5 

25.4 

25.2 

24.9 

24.7 

24.3 

520 

23.0 

23.6 

23.9 

24.3 

24.7 

24.9 

25.2 

25.4 

25.4 

25.3 

25.2 

25.1 

24.8 

540 

2°  ° 

22.G 

23.2 

23.6 

24.0 

24.4 

24.6 

24.9 

25.1 

25.0 

25.1 

25.1 

25.0 

560 

2L3 

21.7 

22.2 

22.8 

23.2 

23.7 

24.0 

24.3 

24.6 

24.7 

24.8 

24.9 

24.9 

580 

20.3 

20.8 

21.3 

21.8 

22.3 

22.7 

23.2 

23.7 

23.9 

24.1 

24.4 

24.6 

24.7 

600 

19.4 

19.9 

20.4 

20.8 

21.4 

21.9 

22.2 

22.7 

23.1 

23.4 

23.7 

24.1 

24.3 

620 

18.5 

19.0 

19.5 

20.1 

20.5 

20.9 

21.4 

21.8 

22.2 

22.6 

22.9 

23.3 

23.6 

640 

17.9 

18.3 

18.7 

19.2 

19.7 

20.1 

20.5 

22.0 

21.3 

21.7 

22.1 

22.5 

22.8 

660 

17.3 

17.6 

18.1 

18.5 

18.9 

19.4 

19.6 

20.1 

20.5 

20.7 

21.2 

21.7 

22.0 

680 

16.8 

17.1 

17.4 

17.8 

18.2 

18.6 

18.9 

19.4 

19.7 

20.1 

20.4 

207 

21.2 

700 

16.4 

16.7 

16.9 

17.3 

17.7 

18.0 

18.3 

18.7 

18.9 

19.2 

19.6 

20.0 

20.3 

720 

16.1 

16.3 

16.5 

16.9 

17.2 

17.6 

17.8 

18.0 

18.3 

18.5 

18.7 

193 

19.5 

740 

15.7 

16.0 

16.2 

16.5 

16.7 

17.0 

17.3 

17.6 

17.8 

17.9 

18.1 

185 

18.8 

760 

15.4 

15.7 

16.0 

16.1 

16.4 

16.6 

16.7 

17.2 

17.4 

17.4 

17.8 

180 

18.2 

780 

14.9 

15.3 

15.6 

15.9 

16.1 

16.3 

16.5 

16.7 

16.9 

17.1 

17.3 

17.6 

17.7 

800 

14.5 

14.7 

15.2 

15.5 

15.8 

15.9 

16.2 

16.5 

16.6 

16.8 

16.9 

17.1 

17.3 

820 

14.0 

14.4 

14.7 

15.1 

15.4 

15.7 

15.8 

16.1 

16.3 

16.4 

16.6 

16.9 

17.0 

840 

13.4 

13.7 

14.1 

14.5 

15.1 

15.4 

15.4 

15.8 

15.9 

16.1 

16.2 

16.6 

16.7 

860 

12.6 

13.1 

13.5 

13.9 

14.3 

14.8 

15.2 

15.5 

15.6 

15.8 

16.0 

16.3 

16.4 

880 

11.8 

12.3 

12.8 

13.3 

13.7 

14.1 

14.5 

15.0 

15.3 

15.4 

15.6 

15.9 

16.1 

900 

10.8 

11.3 

11.9 

12.4 

13.0 

13.4 

13.7 

14.2 

14.7 

15.0 

15.2 

15.5 

15.7 

920 

9.9 

10.3 

10.8 

11.4 

12.0 

12.5 

12.9 

13.4 

14.0 

14.3 

14.7 

15.0 

15.3 

940 

8.9 

9.4 

9.9 

10.4 

11.0 

11.6 

12.1 

12.5 

13.0 

13.6 

13.9 

14.4 

14.7» 

960 

8.0 

8.3 

8.8 

94 

10.0 

10.6 

11.1 

11.7 

12.2 

12.5 

13.1 

13.7 

14.1 

980 

7.3 

7.6 

7.9 

8.4 

8.9 

9.5 

9.9 

10.5 

11.1 

11.6 

12.1 

12.8 

13.3  i 

1000 

6.5 

6.8 

7.2 

7.5 

8.0 

8.4 

8.8 

9.5 

10.0 

10.5 

11.0 

11.6 

12.4 

750 

760 

770 

780 

790 

800 

810 

820 

830 

840 

850 

860 

870 

46 


TABLE  XXXII. 


Perturbations  produced  by  Jupiter. 

Arguments  II.  and  V. 

V. 


II. 

870 

880 

890 

900 

910 

920 

930 

940 

950 

960 

970 

980 

990 

1000 

0 

12.4 

12.9 

13.2 

13.6 

13.9 

14.2 

14.4 

14.8 

15.0 

15.1 

15.1 

15.2 

15.2 

15.3 

20 

11.1 

11.7 

12.2 

12.7 

13.2 

13.6 

13.8 

14.1 

14.4 

14.7 

14.8 

15.0 

14.9 

14.9 

40 

10.0 

10.5 

11.1 

11.7 

12.3 

12.6 

13.0 

13.4 

13.7 

14.1 

14.3 

14.6 

14.7 

14.7 

60 

8.8 

9.4 

9.9 

10.6 

11.2 

11.8 

12.1 

12.6 

12.9 

13.3 

13.6 

13.9 

14.2 

14.4 

80 

7.8 

8.3 

8.7 

9.3 

10.0 

10.5 

11.1 

11.6 

12.1 

12.5 

12.8 

13.2 

13.5 

13.8 

100 

6.8 

7.2 

7.6 

8.1 

8.6 

9.4 

9.9 

105 

10.9 

11.4 

12.0 

12.4 

12.8 

13.2 

120 

5.8 

6.1 

6.6 

7.1 

7.6 

8.1 

8.7 

9.4 

9.9 

10.4 

10.8 

11.4 

11.8 

12.3 

140 

5.1 

5.3 

5.6 

6.0 

6.5 

7.0 

7.5 

8.2 

8.7 

9.3 

9.7 

10.3 

10.8 

11.3 

160 

4.7 

4.8 

4.8 

5.2 

5.6 

5.9 

6.3 

6.8 

7.4 

8.0 

8.6 

9.2 

9.7 

10.2 

180 

4.5 

4.5 

4.4 

4.5 

4.8 

5.1 

5.4 

5.8 

6.2 

6.9 

7.4 

8.0 

3.4 

9.1 

200 

4.7 

4.5 

4.2 

4.2 

4.2 

4.4 

4.6 

5.0 

5.3 

5.7 

6.3 

6.9 

7.4 

7.8 

220 

5.2 

4.7 

4.3 

4.2 

4.1 

4.1 

4.0 

4.3 

4.5 

4.8 

5.1 

5.7 

6.2 

6.8 

240 

5.9 

5.3 

4.7 

4.3 

4.1 

4.0 

3.8 

3.9 

4.0 

4.2 

4.3 

4.7 

5.2 

5.7 

260 

6.9 

6.1 

5.4 

4.9 

4.4 

4.1 

3.8 

3.7 

3.6 

3.7 

3.8 

4.1 

4.3 

4.9 

280 

8.0 

7.2 

6.3 

5.7 

5.2 

4.6 

4.1 

3.8 

3.5 

3.5 

3.5 

3.6 

3.7 

3.9 

300 

9.4 

8.5 

7.5 

6.8 

6.1 

5.4 

4.7 

4.3 

3.9 

3.6 

3.3 

3.3 

3.3 

3.4 

320 

11.2 

10.1 

9.1 

8.1 

7.3 

6.5 

5.7 

5.0 

4.4 

4.0 

3.6 

3.4 

3.2 

3.2 

340 

12.9 

11.8 

10.7 

9.6 

8.7 

7.7 

6.8 

6.0 

5.2 

4.6 

4.1 

3.7 

3.4 

3.2 

360 

14,7 

13.4 

12.3 

11.1 

10.1 

9.2 

8.3 

7.4 

6.4 

5.7 

4.9 

4.3 

3.8 

3.5 

380 

16.5 

15.4 

14.2 

13.0 

11.8 

10.8 

9.7 

8.7 

7.8 

6.9 

6.1 

5.4 

4.6 

4.1 

400 

18.2 

17.2 

16.0 

14.9 

13.8 

12.4 

11.4 

10.4 

9.3 

8.3 

7.3 

6.4 

5.6 

5.0 

420 

19.8 

18.8 

17.7 

16.7 

15.5 

14.4 

13.1 

11.9 

10.9 

9.8 

8.8 

8.0 

6.9 

6.1 

440 

21.2 

20.3 

19.3 

18.3 

17.3 

16.2 

14.9 

13.8 

12.7 

11.5 

10.5 

9.5 

8.4 

7.5 

460 

22.5 

21.6 

20.6 

19.7 

18.9 

17.9 

16.7 

15.6 

14.3 

13.3 

12.2 

10.9 

10.0 

9.0 

480 

235 

22.7 

22.0 

21.1 

20.2 

19.3 

18.2 

17.3 

16.2 

15.0 

13.8 

12-8 

11.6 

10.5 

500 

24.3 

23.8 

23.0 

22.3 

21.6 

20.7 

19.7 

18.8 

17.8 

16.7 

15.4 

14.5 

13.4 

12.3 

520 

24.8 

24.3 

23.7 

23.2 

22.7 

21.9 

21.1 

20.2 

19.2 

18.3 

17.2 

16.1 

15.0 

14.0 

540 

25.0 

24.8 

24.3 

23.9 

23.4 

22.8 

22.1 

21.3 

20.6 

19.7 

18.7 

17.6 

16.6 

15.6 

560 

24.9 

24.8 

24.7 

24-4 

24.0 

23.6 

22.9 

22.4 

21.0 

20.8 

20.0 

19.1 

18.2 

17.1 

580 

24.7 

24.7 

24.6 

24.5 

24.3 

23.9 

23.5 

23.1 

22.5 

21.9 

21.1 

20.3 

19.5 

18.6 

600 

24.3 

24.3 

24.3 

24.3 

24.3 

24.1 

23.8 

23.5 

23.0 

22.5 

22.0 

21.4 

20.6 

198 

620 

23.6 

23.7 

23.9 

24.0 

24.1 

24.1 

23.9 

23.7 

23.4 

23.1 

22.6 

22.1 

21.4 

20.8 

640 

22.8 

23.1 

23.2 

23.4 

23.6 

23.7 

23.8 

23.7 

23.5 

23.2 

22.9 

22.6 

22.1 

21.6 

660 

22.0 

22.3 

22.5 

22.8 

23.0 

23.2 

23.2 

23.3 

23.2 

23.1 

23.0 

22.8 

22.5 

22.1 

680 

21.2 

21.5 

21.7 

22.0 

22.3 

22.5 

22.6 

22.8 

22.9 

22.9 

22.8 

22.7 

22.7 

22.3 

700 

20.3 

20.7 

20.9 

21.2 

21.5 

21.7 

21.9 

22.2 

22.3 

22.5 

22.5 

22.5 

22.4 

22.2 

720 

19.5 

19.8 

20.1 

20.4 

20.8 

21.1 

21.2 

21.4 

21.6 

21.8 

21.9 

22.0 

22.0 

22.0 

740 

18.8 

19.0 

19.2 

19.6 

19.9 

20.2 

20.5 

20.7 

20.9 

21.1 

21.2 

21.5 

21.5 

21.6 

760 

18.2 

18.5 

18.4 

18.8 

19.1 

1.9.4 

19.6 

19.9 

20.1 

20.3 

20.5 

20.8 

21.0 

21.2 

780 

17.7 

17.8 

18.0 

18.1 

18.4 

18.7 

18.8 

19.1 

19.3 

19.5 

19.7 

20.0 

20.2 

20.4 

800 

17.3 

17.4 

17.4 

17.7 

17.9 

18.0 

18.1 

18.4 

18.6 

18.9 

18.9 

19.1 

19.4 

19.6 

820 

17.0 

17.2 

17.2 

17.2 

17.4 

17.4 

17.6 

17.8 

17.8 

18.1 

18.3 

18.5 

18.6 

18.8 

840 

16.7 

16.8 

16.8 

16.9 

17.2 

17.2 

17.1 

17.1 

17.3 

17.4 

17.5 

17.8 

17.9 

18.1 

860 

16.4 

16.5 

16.5 

16.6 

16.6 

16.7 

16.8 

16.9 

16.9 

17.0 

17.0 

17.1 

17.2 

17.4 

880 

16.1 

16.3 

16.3 

16.5 

16.5 

16.5 

16.6 

16.6 

16.6 

16.6 

16.6 

16.7 

16.7 

16.9 

900 

15.7 

15.9 

16.1 

16.2 

16.3 

16.4 

16.3 

16.3 

16.2 

16.2 

16.2 

16.3 

16.3 

16.3 

920 

15.3 

15.5 

15.6 

15.9 

16.0 

16.1 

16.1 

16.1 

16.0 

16.1 

16.1 

16.1 

16.0 

16.0 

940 

14.7 

15.9 

15.2 

15.4 

15.7 

15.8 

15.8 

16.0 

15.9 

15.9 

15.9 

15.8 

15.7 

15.8 

960 

14.1 

14.3 

14.5 

14.8 

15.2 

15.5 

15.5 

15.7 

15.7 

15.7 

15.6 

15.6 

15.5 

15.5 

980 

13.3 

12.7 

13.9 

14.2 

14.5 

14.8 

15.1 

15.3 

15.4 

15.5 

15.4 

15.4 

15.4 

15.3 

1000 

12.4 

12.9 

13.2 

13.6 

13.9 

14.2 

14.4 

14.8 

15.0 

15.1 

15.1 

15.2 

15.2 

15.3 

870 

880 

890 

900 

910 

920 

930 

940 

950 

960 

970 

980 

990 

1000 

TABLE  XXXIII. 
Perturbations  produced  by  Saturn. 

Arguments  II  and  VII. 
VII. 


47 


II 

0 

100 

200 

300 

400 

500 

600 

700 

800 

900 

1000 

0 

1.2 

1.5 

1.4 

1.0 

0.7 

0.6 

0.5 

0.5 

0.4 

0.8 

1.2 

100 

0.9 

1.2 

1.3 

1.1 

0.9 

0.8 

0.7 

0.7 

0.6 

0.7 

0.9 

200 

0.7 

0.9 

1.0 

1.1 

1.0 

0.9 

0.8 

0.8 

0.9 

0.8 

0.7 

300 

0.9 

0.8 

0.7 

0.8 

0.9 

1.0 

1.0 

1.0 

1.0 

.0 

0.9 

400 

.0 

0.9 

0.6 

0.4 

0.6 

0.9 

1.0 

1.1 

1.1 

.1 

1.0 

500 

.1 

1.0 

0.8 

0.4 

0.2 

0.5 

1.0 

1.3 

1.3 

.2 

1.1 

600 

.2 

1.1 

0.9 

0.6 

0.2 

0.2 

0.5 

1.1 

1.5 

.5 

1.2 

700 

.4 

1.1 

1.0 

0.8 

0.4 

0.1 

0.3 

0.8 

1.4 

.7 

1.4 

800 

.6 

1.3 

1.0 

0.8 

0.6 

0.4 

0.1 

0.3 

1.0 

.6 

1.6 

900 

.5 

1.4 

1.1 

0.9 

0.7 

0.6 

0.3 

0.2 

0.6 

.2 

1.5 

1000 

.2 

1.5 

1.4 

1.0 

0.7 

0.6 

0.5 

0.5 

0.4 

0.8 

1.2 

Constant,  l."0 


TABLE  XXXIV. 

Variable  Part  of  Sun's  Aberration. 
Argument,  Sun's  Mean  Anomaly. 


0* 

I* 

Us 

III* 

IV* 

V* 

o 

// 

// 

„ 

Z 

// 

„ 

o 

0 

0.0 

0.0 

0.1 

0.3 

0.5 

0.6 

30 

3 

0.0 

0.0 

0.2 

0.3 

0.5 

0.6 

27 

6 

0.0 

0.0 

0.2 

0.3 

0.5 

0.6 

24 

9 

0.0 

0.0 

0.2 

0.3 

0.5 

0.6 

21 

12 

0.0 

0. 

0.2 

0.4 

05 

0.6 

18 

15 

0.0 

0. 

0.2 

0.4 

0.5 

0.6 

15 

18 

0.0 

0. 

0.2 

0.4 

0.5 

0.6 

12 

21 

0.0 

0. 

0.3 

0.4 

0.6 

0.6 

9 

24 

0.0 

0. 

0.3 

0.4 

0.6 

0.6 

6 

27 

0.0 

0.1 

0.3 

0.4 

0.6 

0.6 

3 

30 

0.0 

0.1 

0.3 

0.5 

0.6 

0.6 

0 

XI* 

x* 

IX* 

VIII* 

VII* 

VI* 

Constant,  0."3 


48 


TABLE  XXXV. 
Moorfs  Epochs. 


Years. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11  12 

£| 

1830  * 

00174 

4541 

4461  4638 

9885  0635  5979  9921 

7623 

219 

226,458 

468 

1831 

00103 

1749 

41279381 

2357 

64327040  2378  6487  825 

58  7|1  77 

940 

1832  B 

00032 

8957 

3793!4125 

4829 

22298100;48355351 

432 

948 

897 

413 

1833 

00235 

6816 

4499  9156 

7636 

8399  9219 

76834239 

108 

340 

687 

920 

1834 

00164 

4024 

4164 

3900 

0107 

41960279  01403103 

715 

701 

406 

393 

1835 

00093 

1232 

3830 

8644 

2579 

99931340 

2598 

1967 

321 

061 

125 

866 

183613 

00022 

8441 

3496 

3388 

5051 

5791  2400 

50550831 

928 

422 

845 

339 

1837 

00224 

6299 

4202|8419  7858  1960  3518 

7903  9719 

605 

814 

635 

846 

1838 

00153 

3508 

3868 

3163  0329  7757  4579 

03608583 

211  175 

354 

319 

1839 

00082 

0716 

3534 

7907 

2801 

3555  5639 

2818 

7447 

818  536 

074 

792 

1840  B 

00011 

7925 

3199 

2651  5273 

9352^6700 

5275 

6310 

424  896 

793 

265 

1841 

00213 

5783 

390676828080 

55227818 

8123 

5199 

101  288 

583 

772 

1842 

00142 

2991 

3571  2425  0551 

13198879 

0580 

4062 

707  649 

302 

245 

1843 

00071 

0200 

3237 

7169  3023 

71  16,  9939 

3038 

2926 

314010 

022 

718 

1844  B 

00000 

7408 

2903 

19135495 

2914 

1000 

5495 

1790 

920  371 

741 

191 

1845 

00203 

5266 

3609 

6944*8302 

9083 

2118 

8343 

0678 

597 

763 

531 

698 

1846 

00132 

2475 

3275 

1688  0773 

4880 

3179 

0800  9542 

203 

123 

250 

171 

1847 

00061 

9683 

2941 

6432,3245 

0678 

4239 

3257'  8406 

810 

484 

970 

644 

1848  B 

99990 

6892 

2606 

11765717 

6475 

5300 

5715 

7270 

416 

845 

689 

117 

1849 

00192 

4750 

3312 

6207,8524 

2644 

6418 

8563 

6158 

093 

237 

479 

624 

1850 

00121 

1958 

2978 

0951 

0995 

8442 

7479 

1020 

5022 

700 

597 

199 

097 

1851 

00050 

9167  2644 

5695  3467 

4239 

8539 

3477 

3885 

306 

958 

918 

570 

1852  B 

99979 

6375  2310 

04395939 

0036 

9600 

5935  2749 

913 

319 

637 

043 

1853 

00181 

4233  3016 

5469  8746 

6206 

0718 

8782 

1637 

589 

711 

427 

550 

1854 

00110 

14422681 

0213 

1217 

2003 

1778 

1240  0501 

196 

072 

147 

023 

1855 

00039 

8650  '2347 

4957 

3689 

78012839 

3697 

9365 

802 

432 

866 

496 

1856  B 

99968 

5859  2013 

9701 

6160 

35983899 

61558229 

409 

793 

586 

969 

1857 

00171 

37172719 

4732  8968 

976715018 

90027117 

086 

185 

375 

476 

1858 

00100 

09252385 

9476  1439 

5565 

607S 

14605981 

692 

546 

095 

949 

1859 

00029 

8  134  '2051 

42203911 

1362 

7139 

39174845 

299 

907 

814 

422 

1860  B 

99958  5342  1716 

8964 

6383 

7159 

8199 

6374  3709 

905 

267 

534 

895 

1861 

00160  3200 

2423 

3995 

9190 

3329 

9317 

9222,2597 

581 

659 

323 

402 

1862 

00089  0409 

2088 

8739 

1661 

9126  0378 

1679 

1461 

188 

020 

043 

875 

1863 

00018  7617 

1754 

3483  4133 

4923  1438 

4137  0324 

795 

381 

762 

348 

1864  B 

99947  4826 

1420 

8227  6605 

0721 

2499 

65949188 

401 

742 

482 

821 

1865 

0014912684 

2126 

3257 

9412 

6890 

3617 

9442  8076 

078 

134 

272 

328 

1866 

00078 

9893 

1792 

8001  1883 

2687 

4678  1899  6940 

685 

494 

991 

801 

1867 

00007 

7101 

1457 

2745  4355 

8485 

5738  4357  5804 

291 

855 

711 

274 

1868  B 

99936 

4309  1123 

7489  6827 

4282 

6799  68144668 

898 

216 

431 

747 

1869 

00138 

2168  1829 

2520  9634 

0452 

791796623556 

574 

608 

220 

254 

1870 

00067 

93761495 

7264  2105  6249 

8978!  21  19  2420 

181 

968 

940 

727 

TABLE  XXXV. 
Moon's  Epochs. 


Years. 

14 

15 

16 

17 

18 

19 

20 

21 

2223 

24 

25 

26 

27 

28 

29|30|31 

1830 

921 

392 

230 

588 

462 

523 

536 

52 

6044 

94 

51 

47 

98 

99 

99  8'9  52 

1831 

115 

532 

589 

940 

937 

296 

703 

30 

7041 

65 

53 

94 

48 

24 

2451 

44 

1832  B 

309 

673 

949 

293 

412 

070  870 

07 

81  38 

36 

55 

42  97 

48  49  14  35 

1833 
1834 

602 
796 

844 
984 

345 

704 

688 
040 

913 

388 

845  037 
619  203 

85 
62 

9245 
0342 

07 

77 

61 
63 

9253 
4003 

77 
01 

777727 
0139J18 

1835 

989 

124 

063 

393 

863 

392  370 

39 

1338 

48 

65 

87 

51 

26 

2602 

10 

1 

1836  B 

183 

265  423 

745 

338 

166 

537 

17 

2435 

19 

67 

34 

01 

50 

51  6401 

1837 

476 

436819 

140 

840 

942 

704 

94 

3542 

90 

73 

8558 

79 

792793 

1838 

670 

576  178 

492 

315 

715 

870 

72 

4638 

60 

75 

32^07 

04 

04 

8984 

1839 

864 

716  537 

845 

790 

489 

037 

49 

5635 

31 

77 

8056 

28 

28 

5276 

1840  B 

058 

857 

8971197 

265 

262 

204 

26 

6732 

02 

79 

27 

06 

53 

52 

1467 

1841 

351 

028 

293592 

766 

038 

371 

04 

7839 

73 

85 

77 

62 

81 

81 

77 

59 

1842 

544 

168 

652944 

241 

811 

537 

81 

89!35 

43 

87 

25 

12 

0606 

40 

51 

1843 

738 

308 

012  297 

716 

585 

704 

58 

9932 

14 

89 

72 

61 

3031 

02 

42 

1844  B 

932 

449 

371 

649 

191 

358 

871 

36 

1029 

85 

01 

19 

10 

55 

55 

65 

34 

1845 

225 

620 

767 

044 

692 

134 

038 

13 

21 

36 

56 

97 

70 

67 

84 

83 

27 

26 

1846 

419 

760 

126 

396 

167 

907 

204 

91 

32 

32 

26 

99 

17 

16 

08 

08 

90 

17 

1847 

613 

901 

486  749 

643 

681 

371 

68 

42 

29 

97 

01 

65 

6533 

33 

52 

09 

1848  B 

806 

041 

845  ' 

101 

118 

454 

538 

45 

53 

26 

68 

03 

12 

15  5758 

15 

00 

1849 

099 

212 

241  : 

496 

619 

230 

705 

23 

64 

33 

39 

09 

03 

71  86  86 

77 

92 

1850 

293 

352 

600  848 

094 

003 

871 

00 

75 

29 

09 

10 

10 

20  10  10 

40 

83 

I 

1851 

487 

493 

960 

201 

569 

777 

038 

78 

85 

26 

80 

12 

57 

70  35!35 

02 

75 

1852  B 

681 

633 

319 

553 

044 

550 

205 

55 

96 

23 

51 

14 

04 

19  59  60 

65 

66 

1853 

974 

«04 

715 

948 

545 

326 

372 

33 

07 

30 

22 

20 

55 

7688 

88 

28 

58 

1854 

168 

944 

074 

300 

020 

099 

539 

10 

18 

26 

93 

22 

03 

25  12 

12 

90 

50 

1855 

361 

085 

434653 

495 

873 

705 

87 

28 

23 

63 

24 

50 

7437 

37 

53 

41 

1 

1856  B 

555 

225 

793005 

970 

64ft 

872 

65 

39 

2034 

26 

97 

23  61 

62 

15 

33 

1857 

848 

396 

189  400 

471 

422 

039 

42 

50 

27105 

32 

48 

80  90 

90 

78  24 

1858 

042 

537 

548 

752 

947 

195 

206 

20 

61 

2476 

34 

95 

29  15 

15 

40 

16 

1859 

236 

677 

908 

105 

422 

969 

372 

97 

71 

2046 

36 

42 

7939 

40 

0307 

1860  B 

430 

817 

267  457 

897 

742 

539 

74 

82 

1717 

38 

89 

28  64  64 

6599 

i 

1861 

723 

988 

663 

852 

398 

518 

706 

52 

93 

2488 

44 

41 

84  92  92 

28 

91 

1862 

916 

129 

022 

204 

873 

291 

873 

29 

04 

20'  60 

46 

88 

34  17 

17 

91 

82 

1863 

110 

269 

382 

557 

348 

065 

039 

06 

14 

1729 

48 

35 

8241 

42 

53 

74 

1864  B 

304 

409 

741 

909 

823 

838 

206 

84 

25 

1400 

50 

82 

32  66  66 

16 

65 

1865 

597 

580 

137304 

324 

614 

373 

61 

36 

21  71 

56 

33 

89  95  94 

78 

57 

1866 

791 

721 

496 

657 

799 

387 

540 

39 

47 

1742 

58 

80 

38  1919 

41 

49 

1867 

985 

861 

856  009  274 

161 

707 

16 

57 

1412 

60 

28 

87  44J44 

03 

40 

1868  B 

178 

001 

215362 

749 

934 

873 

93 

68 

1183 

62 

75 

37  68  69 

66 

32 

1869 

471 

172 

611; 

756 

251 

710 

040 

71 

79 

1854 

68 

26 

93  97  97 

28 

23 

1870 

665 

313 

970 

109 

726J483 

207148 

90 

1526 

69 

73 

43  21  21 

91 

15 

50 


TABLE  XXXV. 
Moon's  Epochs. 


Years. 

Evection. 

Anomaly. 

Variation. 

Longitude. 

,     o     /     // 

s      o      f      /' 

8       0         /        " 

9        0        /          " 

1830 

5   17     4   12 

11  24  31     4.5 

2   13     2  39 

11   22  55  37.7 

1831 

11     7  35  41 

2  23    14  24.6 

6   22  40     4 

4     2   18  42.8 

1832  B 

4  28     7   11 

5  21   57  44.4 

11     2   17  28 

8   11   41   48.0 

1833 

10  29  57  40 

9     3  44  58.5 

3  24     6  21 

1     4   15  28.4 

1834 

4  20  29   11 

0     2  28   18.5 

8     3  43  45 

5   13  38   33.6 

1835 

10   11     0  40 

3     1    U   38.6 

0   13  21    10 

9  23     1   38.8 

1836  B 

4     1   32     9 

5  29   54  58.7 

4  22  58  34 

2     2  24  44.0 

1837 

10     3  22  39 

9   11   42   12.8 

9   14  47  27 

6  24  58   24.5 

/1838 

3  23  54     9 

0   10  25  32.9 

1   24  24  51 

11     4  21   29.8 

1839 

9    14  25  38 

39s  53.1 

6     4     2   16 

3   13  44  35.0 

1840  B 

3     4  57     8 

6     7  52   13.2 

10   13  39  42 

7  23     7  40.4 

1841 

9     6  47  37 

9    19  39  27.5 

3     5  28  33 

0   15  41    20.9 

1842 

2  27  19     7 

0   18  22  47.6 

7  15     5  58 

4  25     4  26.2 

1843 

8   17  50  37 

3    17     6     7.9 

11   24  43  23 

9     4  27  31.6 

1844  B 

2     8  22     7 

6    15  49   28.1 

4     4  20  48 

1    13   50  37.0 

1845 

8   10  12  36 

9   27  36  42.5 

8   26     9  40 

6     6  24   17.5 

1846 

2     0  44     6 

0   26  20     2.8 

1     5  47     5 

10   15  47  23.0 

1847 

7  21   15  35 

3   25     3  23.2 

5   15  24  30 

2  25   10  28.3 

1848  B 

1   11  47     5 

6   23  46  43.5 

9  25     1   55 

7     4  33  33.7 

1849 

7  13  37  35 

10     5  33  57.9 

2   16  50  47 

11   27     7   14.5 

1850 

I     4     9     4 

1     4  17   18.3 

6  26  28   12 

4     6   30   19.9 

1851 

6  24  40  35 

430  38.6 

11     6     5  37 

8    15  53   25.4 

1852  B 

0   15   12     5 

7     1   43  59.2 

3   15  43     3 

0  25   16  31.0 

1853 

6   17     2  34 

10   13  31    13.7 

8     7  31   54 

5   17  50   11.6 

1854 

0     7  34     4 

1    12   14  34.1 

0   17     9  20 

9  27   13   17.2 

1855 

5  28     5  33 

4   10  57  54.7 

4  26  46  44 

2     6  36  22.7 

1856  B 

11    18  37     3 

7     9  41    15.2 

9     6  24  10 

6   15  59  28.2 

1857 

5  20  27  33 

10   21   28   29.8 

1   28   13     2 

11     8  33     9.1 

1858 

11    10  59     2 

1   20   n    50.3 

6     7  50  27 

3   17  56   14.6 

1859 

5     1  30  33 

4   18  55   10.9 

10    17  27  53 

7  27   19  20.1 

1860  B 

10  22     2     3 

7  17  38  31.4 

2  27     5   18 

0     6  42  25.8 

1861 

4  23  52  32 

10  29  25  46.1 

7   18  54   10 

4  29   16     6.6 

1862 

10   14  24     2 

1   28     9     6.6 

11   28  31   35 

9     8  39   12.2 

1863 

4     4  55  32 

4  26   52  27.3 

4891 

1    18     2   17.9 

1864  B 

9  25  27     2 

7  25   35  48.0 

8   17  46  25 

5  27  25  23.5 

1865 

3  27  17  31 

11     7  23     2.7 

1     9  35   is 

10   19  59     4.3 

1866 

9   17  49     2 

266  23.3 

5   19   12  43 

2  29  22   10.1 

1867 

3     8  20  31 

5     4  49  44.0 

9  28  50     9 

7     8  45    15.7 

1868  B 

8  28  62     2 

8     3  33     4.7 

2     8  27  34 

11    18     8   21.4 

1869 

3     0  42  83 

11    15   20  19.6 

7     0  16  26 

4   10  42     23 

1870 

«  ai  u    2 

2  14     3  40.3 

11     9  53  51 

8  20     5     8.0 

TABLE  XXXV. 
Moon's  Epochs. 


Years. 

Supp.  of  Node. 

II 

V 

VI 

VII 

VIII 

IX 

X 

XI 

XII 

g   0   '   " 

«   0   " 

1830 

677  11.0 

10  24  46 

498 

502 

900 

904 

427 

062 

025 

433 

1831 

6  26  26  53.3 

215  18 

912 

914 

208 

210 

506 

001 

211 

710 

1832  B 

7  15  46  35.5 

6  550 

326 

327 

516 

516 

586 

940 

397 

986 

1833 

8  5  928.4 

10  731 

774 

779 

852 

856 

702 

885 

624 

297 

1834 

8  24  29  10.7 

128  3 

187 

191 

159 

163 

782 

825 

810 

573 

1835 

9  13  48  53.0 

5  18  35 

601 

603 

467 

469 

861 

764 

996 

850 

1836  B 

10  3  8  35.2 

998 

015 

016 

775 

775 

941 

703 

182 

127 

1837 

102231  28.1 

1  1049 

463 

468 

111 

116 

057 

648 

409 

437 

1838 

11  1151  10.4 

5  1  21 

876 

880 

419 

423 

137 

588 

595 

714 

1839 

0  1  10  52.6 

8  21  53 

290 

292 

726 

729 

217 

527 

781 

991 

1840  B 

0  20  30  34.9 

0  1225 

704 

705 

034 

035 

296 

466 

967 

268 

1841 

1  9  53  27.7 

4  14  6 

152 

157 

370 

375 

412 

411 

194 

578 

1842 

1  29  13  10.0 

8  438 

566 

569 

678 

682 

492 

350 

380 

855 

1843 

2  18  32  52.2 

11  25  10 

980 

980 

986 

988 

572 

290 

566 

131 

1844  B 

3  7  52  34.5 

3  1542 

393 

394 

293 

294 

651 

229 

752 

408 

1845 

3  27  15  27.4 

7  1723 

840 

846 

629 

634 

767 

174 

979 

718 

1846 

41635  9.6 

11  7*5 

254 

258 

937 

941 

847 

113 

165 

995 

1847 

5  55451.8 

22827 

668 

670 

245 

247 

927 

053 

351 

272 

1848  B 

525  1434.1 

6  18  59 

082 

083 

553 

553 

006 

992 

537 

549 

1849 

6  14  37  27.0 

10  20  40 

531 

535 

889 

893 

122 

937 

764 

859 

1850 

7  357  9.2 

211  12 

944 

947 

196 

200 

202 

876 

950 

136 

1851 

723  1651.5 

6  144 

358 

359 

504 

506 

282 

816 

136 

413 

1852  B 

8  12  36  33.6 

922  17 

772 

772 

812 

812 

362 

755 

322 

689 

1853 

9  1  59  26.5 

1  2358 

220 

223 

148 

152 

477 

700 

549 

000 

1854 

9  21  19  8.8 

5  1430 

634 

636 

456 

459 

557 

639 

735 

276 

1855 

10  103851.1 

952 

047 

048 

763 

765 

637 

579 

921 

553 

1856  B 

10  29  58  33.3 

02534 

461 

461 

71 

071 

717 

518 

107 

830 

1857 

11  192126.2 

42715 

909 

912 

407 

411 

832 

463 

334 

140 

1858 

0  841  8.4 

8  1747 

323 

325 

715 

718 

912 

402 

520 

417 

1859 

028  050.7 

0  8  19 

736 

737 

023 

024 

992 

342 

706 

694 

1860  B 

1  17  20  32.9 

32851 

150 

150 

330 

330 

072 

281 

892 

971 

1861 

2  64325.8 

8  032 

598 

601 

666 

670 

187 

226 

119 

281 

1862 

226  3  8.0 

11  21  4 

012 

014 

974 

977 

267 

165 

305 

558 

1863 

3  15  22  50.1 

3  1136 

426 

426 

282 

283 

347 

105 

491 

834 

1864  B 

4  44232.3 

728 

839 

839 

590 

589 

427 

044 

677 

111 

1865 

424  525.2 

11  349 

287 

291 

926 

929 

542 

989 

904 

422 

1866 

5  13  25  7.3 

22421 

701 

703 

233 

236 

622 

928 

090 

698 

1867 

6  24449.5 

6  1453 

115 

115 

541 

542 

702 

868 

276 

975 

1868  B 

622  431.7 

10  526 

529 

528 

849 

848 

782 

807 

462 

252 

1869 

7112724.6 

277 

977 

980 

185 

188 

897 

752 

689 

562 

1670 

8  047  6.7 

52739 

390 

392 

493 

495 

977 

691 

875 

839 

TABLE  XXXVI. 

Moon's     Motions  for  Months. 


Months. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

January 

00000 

0000 

0000 

0000 

0000 

0000 

0000 

0000 

0000 

000 

000 

000 

000 

February 

08487 

0146 

2246 

8896 

0402 

1533 

1789 

2099 

0753 

175 

965  184 

059 

Mo,.  V,  5  Com' 

16153 

8343 

1371 

6931 

9797 

1951 

3404 

3027 

1433 

139 

836 

157 

016 

March  \  Bis. 

16427 

8993 

2411 

7218 

0132 

2323 

3462 

3418 

1457 

209 

868 

228 

050 

.,  (  Com. 

24640 

8490 

3616 

5827 

0199 

3484 

5193 

5126 

2186 

314 

801 

342 

076 

April  <  T, 
r    <  Bis. 

24914 

9140 

4657 

6114 

0534 

3856 

5251 

5517 

2210 

384 

832 

412 

110 

,,    (  Com. 

32853 

7986 

4822 

4436 

0265 

4646 

6924 

6835 

2914 

419 

735 

456 

101 

M*y  isis. 

33127 

8636 

5862'4723 

0600 

5018 

6982 

7226 

2938 

489 

766 

526 

135 

T     (  Com. 

41340 

8133 

7067  3332 

0666 

6179 

8713 

8934 

3667 

593 

700 

640 

160 

June  <  g:g 

41614 

8783 

8107 

3619 

1002 

6551 

8771 

93253691 

663 

731 

710 

194 

T  ,    (  Com. 

49554 

7629 

8273 

1942 

0732 

7341 

0444 

0643!4396 

698 

634 

754 

185 

Ju]y  {Bis. 

49828 

8279 

9313 

2228 

1068 

7713 

0502 

10344420 

768 

665 

824 

219 

.    5  Com. 

58041 

7776 

05-18 

0838 

11348874 

2233 

2742  5148 

873 

599 

938 

245 

A°g-  Us. 

58315 

8426 

1558 

1125 

1470 

9246 

2290 

31335173 

943 

630 

009 

279 

0  .   (  Com. 

66528 

7922 

2764 

9734 

1536 

0408 

4021 

48425901 

048 

563 

123 

304 

S*&'  ifiis. 

66802 

8572 

3804 

0021 

1871 

0780 

4079 

523215925 

118 

595 

193 

338 

f*.    <  Com. 

74741 

7419 

3969 

8343 

1602 

1569 

5752 

6550  6630 

152 

497 

237 

329 

UCt.     <  T> 

75015 

8069 

5009 

8630 

1938 

1941 

5810 

6941 

6654 

222 

528 

307 

363 

lyr    (  Com. 

83228 

7565 

6215 

7239 

2004 

3102 

7541 

8649 

7382 

327 

462 

421 

388 

Nov-  }Bis. 

83502 

8215 

7255 

7526 

2339 

3475 

7599 

9040 

7407 

397 

493 

492 

423 

TX    <  Com. 

91442 

7062 

7420 

5848 

2070 

4264 

9272 

03588111 

432 

396 

535 

414 

\  Bis. 

91716 

7712 

8460 

6135 

2405 

4636 

9330 

0749 

8135 

502 

427 

606 

448 

TABLE  XXXVI. 

Moon's  Motions  for  Months. 

Months. 

Erection. 

Anomaly.   |   Variation. 

Longitude. 

January 

0000 

0  0  0  0.0 

000 

0 

0  0  0  0.0 

February 

11  20  48  42 

1  15  0  53.1 

0  17  54 

48 

1  18  28  5.8 

March  J  Com' 

10  7  40  26 

1  20  50  4.2 

11  29  15 

15 

1  27  24  26.6 

n  \  Bis. 

10  18  59  26 

2  3  53  58.2 

0  11  26 

42 

2  10  35  1.6 

...  (  Com. 

9  28  29  8 

3  5  50  57.3 

0  17  10 

3 

3  15  52  32.5 

AP111  }Bis. 

10  9  48  8 

3  18  54  51.2 

0  29  21 

29 

3  29  3  7.5 

u   J  Com- 

9  7  58  51 

4  7  47  56.4 

0  22  53 

24 

4  21  10  3.3 

y   <  Bis. 

9  19  17  50 

4  20  51  50.3 

1  5  4 

50 

5  4  20  38.3 

June  iCom' 

8  28  47  33 

5  22  48  49.4 

1  10  48 

11 

6  9  38  9.1 

•  MlUt    <  •¥}• 

9  10  6  33 

6  5  52  43.4 

1  22  59 

38 

6  22  48  44.1 

T  ,   (  Com. 

8  8  17  16 

6  24  45  48.5 

1  16  31 

32 

7  14  55  39.9 

July  iBis. 

8  19  36  15 

7  7  49  42.5 

1  28  42 

59 

7  28  6  15.0 

(  Com. 

7  29  5  59 

8  9  46  41.6 

2  4  26 

20 

9  3  23  45.8 

VS'  \  Bis. 

8  10  24  58 

8  22  50  35.5 

2  16  37 

47 

9  16  34  20.8 

Sept  $  C?m- 

7  19  54  41 

9  24  47  34.6 

2  22  21 

7 

10  21  51  51.6 

8  1  13  40 

10  7  51  28.6 

3  4  32 

34 

11  5  2  26.7 

Oct   J  Com< 

6  29  24  24 

10  26  44  33.7 

2  28  4 

28 

11  27  9  22.4 

Uct'   \  Bis. 

7  10  43  23 

11  9  48  27.7 

3  10  15 

55 

0  10  19  57.5 

Nov   J  Com' 

6  20  13  6 

0  11  45  26.8 

3  15  59 

16 

1  15  37  28.3 

llOV»     N  -!»• 

7  1  32  5 

0  24  49  20.7 

3  28  10 

43 

1  28  48  3.3 

T*    (  Com. 

5  29  42  49 

1  13  42  25.9 

3  21  42 

37 

2  20  54  59.1 

Dec.  <  T>- 

6  11  1  48 

1  26  46  19.8 

4  3  54 

4 

345  34.1 

TABLE  XXXVI. 
Moon's     Motions  for  Months. 


53 


Months. 

14 

15 

16 

17 

18 

19 

20 

21 

22 

22 

24 

25 

2627 

28 

292 

0  31 

January 

000 

000 

000 

000 

000000 

000 

00 

00 

OC 

00 

30 

00  00 

00 

00  <] 

0  00 

February           f 

074 

946 

135 

304 

805  066 

014  24 

26 

14 

82 

28 

14 

17 

29 

96  C 

507 

Marrh   J  C°m' 

851 

801 

159 

482 

532 

125 

|027 

4550 

96 

57 

13 

18 

12 

46 

821 

0  15 

March  i  Bis. 

950 

831 

196 

524 

558 

127 

027 

4651 

06 

59 

17 

21 

19 

51 

851 

0  15 

.,     (  Com. 
APnl    iBis. 

925 
024 

747 

778 

294 
331 

786 

828 

336 

362 

191 
193 

041 
042 

6877 
6977 

12 

22 

39 

42 

70 

n 

32 

36 

29 
36 

76 
80 

771 
801 

523 
6  23 

\/i          *  Com- 

899  663 

392 

047 

115 

254 

055 

91  02 

15 

19 

H 

43 

38 

01 

702 

1  30 

ivxav      s  Tj* 

•*          (  JJlS. 

999  693 

429 

089 

141  256 

055 

92,03 

26 

22 

)S 

47 

45,05 

732 

1  30 

June     \  C?m< 

973 

609 

527 

351 

920320 

069 

15 

28 

29 

01 

21 

57 

55  31 

652 

638 

073 

639 

563 

393 

946 

322 

069 

15 

29 

40 

04 

25 

61 

62 

35 

682 

638 

T  ,        (  Com. 

948 

525 

625 

613 

699 

384  083 

37 

54 

33 

81 

15 

68 

6456 

583 

1  45 

y      (  Bis. 

047 

555 

661 

655 

725 

386  083 

38 

55 

43 

84 

1:972 

71 

60 

613 

146 

.           C  Com. 

022 

471 

759 

917 

503 

449  097 

61 

80 

47 

64 

82 

81 

85 

533 

653 

Aug.      <  T,. 
0          I  DIS. 

121 

501 

796 

959 

529 

451 

097 

62 

81 

5? 

66 

r786 

88 

90 

563 

653 

Q     .      (  Com. 

096 

417 

894 

221 

308  515 

111 

85 

07 

61 

46 

)097 

97 

15 

494 

261 

oept.    <  T»- 

195 

447 

931 

263 

334517 

111 

85 

08 

71 

49 

)401 

04 

19 

524 

261 

Oct       J  Com- 

071 

333 

992 

483 

087  578 

125 

07 

32 

65 

26 

23  08 

07 

40 

414 

768 

1  Bis. 

170 

363 

029 

525 

113 

581 

126 

08 

33 

75 

28 

J8 

11 

14 

44 

444 

769 

Nov      J  Com' 

145 

279 

127 

787  892 

644 

139 

31 

59 

79 

08 

il 

22 

23 

70 

375 

276 

(  Bis. 

244 

309 

163 

829 

918  646 

140 

32 

60 

89 

11 

)526 

30 

74 

405 

2|76 

Dec       (  Com. 

120 

194 

225 

049 

670  708 

153 

54 

85 

83 

88 

r433 

3395 

295 

784 

219 

225  261 

091; 

696  710 

153 

54 

86 

93 

90 

r9,37 

40^99 

325 

7^84 

TABLE 

XXXVL 

Moon's  Motions  for  Months.                     \  ^. 

Months. 

Supp.  of  Node. 

II 

V 

VI 

VII 

VIII 

IX 

X 

XI 

XII 

«     o     ' 

s 

0 

/ 

January 

0     0     0     0.0 

0 

0     0 

000 

000 

000 

000 

000 

000 

000 

000 

February 

0     1  38  29.7 

11   15  43 

054 

224 

875 

045 

111 

165 

290 

043 

March  i  Com- 

037  27.5 

9  27  59 

007 

330 

666 

989 

114 

313 

455 

984 

\  Bis. 

0     3  10  38.2 

10 

9     8 

041 

369 

694 

023 

150 

319 

496 

018 

April     j°om- 

0     4  45  57.3 

9  13  42 

061 

554 

542 

034 

225 

478 

745 

027 

0     4  49     7.9 

9  24  51    095 

593 

570 

068 

261 

484 

787 

061 

VTav      J  Com- 

0     6  21   16.4 

8  18  15    081 

738 

389 

046 

300 

638 

993 

036 

jviay      <  T>  • 

(   JJlS. 

0     6  24  27.0 

8  29  25 

115 

778 

417 

080 

336 

643 

034 

070 

June     \  Com- 

0     7  59  46.1 

8 

3  58 

136 

962 

264 

091 

411 

802 

282 

079 

082  56.7 

8  15     8 

170 

002 

293 

124 

447 

808 

324 

113 

j  ,         (  Com. 

0     9  35     5.2 

7 

8  32 

156 

147 

112 

103 

486 

962 

531 

088 

July      \  Bis. 

0     9  38  15.9 

7  19  41    190 

186 

140 

136 

522 

967 

572 

122 

Aug.     jCom' 

0  11  13  35.0 

6  24  15    210  371 

987 

147 

597 

126 

820 

131 

0  11   16  45.6 

7 

5  24 

244 

411 

015 

182 

6 

33 

132 

862 

164 

ge  fc      (  Com. 

0  12  52     4.7 

6 

9  58 

265 

595 

862 

193 

708 

291 

110 

173 

ep  '     (  Bis. 

0  12  55  15.4 

6  21     7 

299 

635 

891 

227 

744 

296 

152 

207 

Oct       \  Com' 

0  14  27  23.8 

5  14  32 

285 

780 

710 

204 

783 

451 

358 

182 

Uct'      \  Bis. 

0  14  30  34.4 

5  25  41 

319 

819 

738 

238 

819 

456 

400 

216 

Nov      $  Com- 

0  16     5  53  5 

5 

0  15 

339 

004 

585 

250 

894 

615 

648 

225 

}B;S. 

0  16     9     4.2 

5  11  24 

373 

043 

613 

283 

930 

621 

690 

259 

j^       (  Com. 

0  17  41  12.6 

4 

4  49 

359 

188 

432 

261 

969 

775 

896 

234 

0  17  44  23.3 

4  15  58 

393 

228 

461 

295 

005  |  780 

938 

268 

TABLE  XXXVII. 


Moon's  Motions  for  Days. 


D. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

"1 
13 

1 

00000 

0000 

0000 

0000 

0000 

0000 

0000 

0000 

0000 

000 

000 

000 

000  i 

2 

00274 

0650 

1040 

0287 

0336 

0372 

0058 

0390 

0024 

070 

031 

070 

034; 

3 

00548 

1300 

2080 

0574 

0671 

0744 

0115 

0781 

0049 

140 

062 

141 

068| 

4 

00821 

1950 

3121 

0861 

1007 

1116 

0173 

1171 

0073 

210 

093 

211 

103 

5 

01095 

2600 

4161 

1148 

1342 

1488 

0231 

1561 

0097 

281 

125 

282 

137 

6 

01369 

3249 

5201 

1435 

1678 

1860 

0289 

1952 

0121 

351 

156 

352 

171 

7 

01643 

3899 

6241 

1722 

2013 

2232 

0346 

2342 

0146 

421 

187 

423 

205 

8 

01916 

4549 

7281 

2009 

2349 

2604 

0404 

2732 

0170 

491 

218 

493 

239 

9 

02190 

5199 

8321 

2296 

2684 

2976 

0462 

3122 

0194 

561 

249 

564 

273 

10 

02464 

5849 

9362 

2583 

3020 

3348 

0519 

3513 

0219 

631 

280 

634 

308 

11 

02738 

6499 

0402 

2870 

3355 

3720 

0577 

3903 

0243 

702 

311 

705 

342 

12 

03012 

7149 

1442 

3157 

3691 

4093 

0635 

4293 

0267 

772 

342 

775 

376 

13 

03285 

7799 

2482 

3444 

4026 

4465 

0692 

4684 

0291 

842 

374 

845 

410 

14 

03559 

8449 

3522 

3731 

4362 

4837 

0750 

5074 

03161912 

405 

916 

444 

15 

03833 

9098 

4563 

4018 

4698 

5209 

0808 

5464 

0340  982 

436 

986 

478 

16 

04107 

9748 

5603 

4305 

5033 

5581 

0866 

5854 

0364  052 

467 

057 

513 

17 

04380 

0398 

6643 

4592 

5369 

5953 

0923 

6245 

0389  122 

498 

127 

547 

18 

04654 

1048 

7683 

4878 

5704 

6325 

0981 

6635 

0413 

193 

529 

198 

581 

19 

04928 

1698 

8723 

5165 

6040 

6697 

1039 

7025 

0437 

263 

560 

268 

315 

20 

05202 

2348 

9763 

5452 

6375 

7069 

1096 

7416 

0461 

333 

591 

339 

649 

21 

05476 

2998 

0804 

5739 

6711 

7441 

1154 

7806 

0486 

403 

623 

409 

683 

22 

05749 

3648 

1844 

6026 

7046 

7813 

1212 

8196 

0510 

473 

654 

480 

718 

23 

06023 

4298 

2884 

6313 

7382 

8185 

1269 

8586 

0534 

543 

685 

550 

752 

24 

06297 

4947 

3924 

6600 

7717 

8557 

1327 

8977 

0559 

614 

716 

621 

786 

25 

06571 

5597 

4964 

6887 

8053 

8929 

1385 

9367 

0583 

684 

747 

691 

820 

26 

06844 

6247 

6005 

7174 

8389 

9301 

1443 

9757 

0607 

754 

778 

762 

854 

27 

07118 

6897 

7045 

7461 

8724 

9673 

1500 

0148 

0631 

824 

809 

832 

888 

28 

07392 

7547  8085 

7748 

9060 

0045 

1558 

0538 

0656 

894 

840 

903 

923 

29 

07666 

8  1  47!  9125 

8035 

9395 

0417 

1616  0928 

0680 

964 

872 

973 

957 

30 

07940 

8847^0165 

8322 

9731 

0789 

1673I15T19 

0704 

034 

903 

043 

991  ! 

31 

08213 

9497  1205  8609 

0066 

1161 

1731  [1709 

0729 

105 

934 

114 

025 

TABLE  XXXVII 


55 


Moon's  Motion  for  Days. 


D. 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

1 

000 

000 

000 

000 

000 

000 

000 

00 

00 

00 

00 

00 

00 

00 

00 

00 

00 

00 

2 

099 

031 

037 

042 

026 

002 

000 

01 

01 

10 

03 

04 

04 

07 

04 

03 

00 

00 

3 

198 

061 

073 

084 

052 

004 

001 

02 

02 

20 

05 

08 

07 

14 

08 

06 

00 

00 

4 

297 

092 

110 

126 

078 

006 

001 

02 

03 

30 

08 

12 

11 

21 

13 

09 

01 

01 

5 

397 

122 

146 

168 

104 

008 

002 

03 

03 

41 

11 

16 

15 

28 

17 

12 

01 

01 

6 

496 

153 

183 

210 

130 

Oil 

002 

04 

04 

51 

13 

21 

18 

35 

21 

15 

01 

01 

7 

595 

183 

220  252 

156 

013 

003 

05 

05 

61 

16 

25 

22 

42 

25 

18 

01 

01 

8 

694 

214 

256 

294 

182 

015 

003 

05 

06 

71 

19 

29 

26 

49 

29 

22 

01 

02 

9 

793 

244 

293 

336 

208 

017 

004 

06 

07 

81 

21 

33 

30 

56 

33 

25 

01 

02 

10 

892 

275 

329 

379 

234 

019 

004 

07 

08 

91 

24 

37 

33 

63 

38 

28 

02 

02 

11 

992 

305 

366 

421 

260 

021 

005 

08 

09 

01 

27 

41 

37 

70 

42 

31 

02 

02 

12 

091 

336 

403 

463 

286 

023 

005 

08 

09 

11 

29 

45 

41 

77 

46 

34 

02 

03 

13 

190 

366 

439 

505 

312 

025 

005 

09 

10 

22 

32 

49 

44 

84 

50 

37 

02 

03 

14 

289 

397 

476 

547 

337 

028 

006 

10 

11 

32 

34 

53 

48 

91 

54 

40 

02 

03 

15 

388 

427 

512 

589 

363 

030 

006 

11 

12 

42 

37 

58 

52 

98 

58 

43 

02 

03 

16  487 

458 

549 

631 

389 

032 

007 

11 

13 

52 

40 

62 

55 

05 

63 

46 

03 

04 

17 

587 

488 

586 

6731 

415 

034 

007 

12 

14 

62 

42 

66 

59 

12 

67 

49 

03 

04 

18 

686 

519 

622 

715  441 

036 

008 

13 

14 

72 

45 

70 

63 

19 

71 

52 

03 

04 

19 

785 

549 

659 

757  ,  467 

038 

008 

14 

15 

82 

48 

74 

66 

26 

75 

55 

03 

04 

20 

884 

580 

695 

799  ,  493 

040 

009 

14 

16 

92 

50 

78 

70 

33 

79 

59 

03 

05 

21 

983 

611 

732 

841  l  519 

042 

009 

15 

17 

03 

53 

82 

74 

40 

84 

62 

03 

05 

22 

082 

641 

769 

883  545 

044 

010 

16 

18 

13 

56 

86 

77 

47 

88 

65 

04 

05 

23 

182 

672 

805 

925  571 

047 

010 

17 

19 

23 

58 

90 

81 

54 

92 

68 

04 

05 

24 

281 

702 

842 

967  j  597 

049 

Oil 

17 

20 

33 

61 

95 

85 

61 

96 

71 

04 

06 

25 

380 

733 

878 

009 

623 

051 

Oil 

18 

20 

43 

64 

99 

89 

68 

00 

74 

04 

06. 

26 

479 

763 

915 

052 

649 

053 

Oil 

19 

21 

53 

66 

03 

92 

75 

04 

77 

04 

06 

27 

578 

794 

952 

094  675 

055 

012 

20 

22 

63 

69 

07 

90 

82 

09 

80 

04 

06 

28 

677 

824 

988 

136  701 

057 

012 

20 

23 

73 

72 

11 

00 

89 

13 

83 

05 

06 

29 

777 

855 

025 

178j727 

059 

013! 

21 

24 

84 

74 

15 

03 

96 

17 

86 

05 

07 

30 

876 

885 

061 

220  ;  753 

061 

013 

22 

25 

94 

77 

19 

07 

03 

21 

89 

05 

07 

31 

975 

916 

098 

262  |  779 

064'  014 

23 

26 

04 

80 

23 

11 

10 

25 

92 

05 

07 

56 


TABLE  XXXVII. 


Moon's  Motions  for  Days. 


D. 

Evection. 

Anomaly. 

Variation. 

M.  Longitude. 

*    °     ' 

8        0        '       " 

8       °        '      " 

8        °        '       " 

1 

0000 

0    0    0    00 

0000 

0    0    0    00 

2 

0  11  18  59 

0  13    3  54.0 

0  12  11  27 

0  13  10  35.0 

3 

0  22  37  59 

0  26     7  47.9 

0  24  22  53 

0  26  21  10.1 

4 

1     3  56  58 

1     9  11  41.9 

1     6  34  20 

1     9  31  45.1 

5 

1  15  15  58 

1  22  15  35.9 

1  18  45  47 

1  22  42  20.1 

6 

1  26  34  57 

2    5  19  29.8 

2    0  57  13 

2    5  52  55.1 

7 

2    7  53  57 

2  18  23  23.8 

2  13    8  40 

2  19    3  30.2 

8 

2  19  12  56 

3     1  27  17.8 

2  25  20    7 

3     2  14    5.2 

9 

3    0  31  55 

3  14  31  11.7 

3    7  31  34 

3  15  24  40.2 

10 

3  11  50  55 

3  27  35    5.7 

3  19  43    0 

3  28  35  15.2 

11 

3  23    9  54 

4  10  38  59.7 

4     1  54  27 

4  11  45  50.3 

12 

4    4  28  54 

4  23  42  53.7 

4  14    5  54 

4  24  56  25.3 

13 

4  15  47  53 

5    6  46  47.6 

4  26  17  20 

5    8     7    0.3 

14 

4  27    6  53 

5  19  50  41.6 

5    8  28  47 

5  21  17  35.4 

15 

5    8  25  52 

6     2  54  35.6 

5  20  40  14 

6    4  28  10.4 

16 

5  19  44  51 

6  15  58  29.5 

6    2  51  40 

6  17  38  45.4 

17 

6     1     3  51 

6  29    2  23.5 

6  15    3    7 

7    0  49  20.4 

18 

6  12  22  50 

7  12    6  17.5 

6  27  14  34 

7  13  59  55.5 

19 

6  23  41  50 

7  25  10  11.4 

7    9  26     1 

7  27  10  30.5 

20 

7    5    0  49 

8    8  14    5.4 

7  21  37  27 

8  10  21     5.5 

21 

7  16  19  49 

8  21  17  59.4 

8     3  48  54 

8  23  31  40.5 

22 

7  27  38  48 

9    4  21  53.4 

8  16    0  21 

9    6  42  15.6 

23 

8    8  57  47 

9  17  25  47.3 

8  28  11  47 

9  19  52  50.6 

24 

8  20  16  47 

10    0  29  41.3 

9  10  23  14 

10    3    3  25.6 

25 

9    1  35  46 

10  13  33  35.3 

9  22  34  41 

10  16  14    0.7 

26 

9  12  54  46 

10  26  37  29.2 

10    4  46    7 

10  29  24  35.7 

27 

9  24  13  45 

11     9  41  23.2 

10  16  57  34 

11  12  35  10.7 

28 

10    5  32  45 

11  22  45  17.2 

10  29     9     1 

11  25  45  45.7 

29 

10  16  51  44 

0    5  49  11.1 

11  11  20  28 

0    8  56  20.8 

30 

10  28  10  43 

0  18  53    5.1 

11  23  31  54 

0  22    6  55.8 

31 

11    9  29  43 

1     1  56  59.1 

0    5  43  21 

1     6  17  30.8 

TABLE.   XXXVII. 


57 


Moon's  Motions  for  Days. 


D. 

Supp.  of  Node. 

II 

V 

VI 

VII 

VIII 

IX 

X 

XI 

XII 

8  °    '   " 

*  °  ' 

1 

00  0  0.0 

000 

000 

000 

000 

000 

000 

000 

000 

000 

2 

003  10.6 

0  11  9 

034 

039 

028 

034 

036 

005 

042 

034 

3 

006  21.3 

0  22  18 

068 

079 

056 

067 

072 

Oil 

083 

067 

4 

009  31.9 

1  3  27 

102 

118 

085 

101 

108 

016 

125 

101 

5 

0  0  12  42.5 

1  14  37 

136 

158 

113 

135 

143 

021 

166 

135 

6 

0  0  15  53.2 

1  25  46 

170 

197 

141 

169 

179 

027 

208 

168 

7 

0  0  19  3.8 

2  6  55 

204 

237 

169 

202 

215 

032 

250 

202 

8 

0  0  22  14.5 

2  18  4 

238 

276 

198 

236 

251 

037 

291 

235 

9 

0  0  25  25.1 

2  29  13 

272 

316 

226 

270 

287 

043 

333 

269 

10 

0  0  28  35.7 

3  10  22 

306 

355 

254 

303 

323 

048 

374 

303 

11 

0  0  31  46.4 

3  21  31 

340 

395 

282 

337 

358 

053 

416 

336 

12 

0  0  34  57.0 

4  2  40 

374 

434 

311 

371 

394 

058 

458 

370 

13 

0  0  38  7.6 

4  13  50 

408 

474 

339 

405 

430 

064 

499 

404 

14 

0  0  41  18.3 

4  24  59 

442 

513 

367 

438 

466 

069 

541 

437 

15 

0  0  44  28.9 

568 

476 

553 

395 

472 

502 

074 

583 

471 

16 

0  0  47  39.5 

5  17  17 

510 

592 

424 

506 

538 

080 

624 

505 

17 

0  0  50  50.2 

5  28  26 

544 

632 

452 

539 

573 

085 

666 

538 

18 

0  0  54  0.8 

6  9  35 

578 

671 

480 

573 

609 

090 

707 

572 

19 

0  0  57  11.5 

6  20  44 

612 

711 

508 

607 

645 

096 

749 

605 

20 

010  22.1 

7  1  53 

646 

750 

537 

641 

681* 

101 

791 

639 

21 

013  32.7 

7  13  3 

680 

790 

565 

674 

717 

106 

832 

673 

22 

0  1  6  43.4 

7  24  12 

714 

829 

593 

708 

753 

112 

874 

706 

23 

0    9  54.0 

8  5  21 

748 

869 

621 

742 

788 

117 

915 

740 

24 

0   13  4.6 

8  16  30 

782 

908 

650 

775 

824 

122 

957 

774 

25 

0   16  15.3 

8  27  39 

816 

948 

678 

809 

860 

128 

999 

807 

26 

0   19  25.9 

9  8  48 

850 

987 

706 

843 

896 

133 

040 

841 

27 

0   82  36.5 

9  19  57 

884 

027 

734 

877 

932 

138 

082 

875 

28 

0   25  47.2 

10  1  6 

918 

066 

762 

910 

968 

143 

123 

908 

29 

0   28  57.8 

10  12  16 

952 

106 

791 

944 

003 

149 

165 

942 

30 

0  1  32  8.5 

10  23  25 

986 

145 

819 

978 

039 

154 

207 

975 

31 

0  1  35  19.1 

11  4  34 

020  | 

185 

847 

Oil 

075 

159 

248 

009  / 

58 


TABLE  XXXVIII. 

Maoris  Motions  for  Hours. 


H. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

1 

11 

27 

43 

12 

14 

16 

2 

16 

1 

3 

1 

3 

1 

2 

23 

54 

87 

24 

28 

31 

5 

33 

2 

6 

3 

6 

3 

3 

34 

81 

130 

36 

42 

47 

7 

49 

3 

9 

4 

9 

4 

4 

46 

108 

173 

48 

56 

62 

10 

65 

4 

12 

5 

12 

6 

5 

57 

135 

217 

60 

70 

78 

12 

81 

5 

15 

6 

15 

7 

6 

68 

162 

260 

72 

84 

93 

14 

98 

6 

18 

8 

18 

9 

7 

80 

190 

303 

84 

98 

109 

17 

114 

7 

20 

9 

20 

10 

8 

91 

217 

347 

96 

112 

124 

19 

130 

8 

23 

10 

23 

11 

9 

103 

244 

390 

108 

126 

140 

22 

146 

9 

26 

12 

26 

13 

10 

114 

271 

433 

120 

140 

155 

24 

163 

10 

29 

13 

29 

14 

11 

125 

298 

477 

131 

154 

171 

26 

179 

11 

32 

14 

00 

16 

12 

137 

325 

520 

143 

168 

186 

29 

195 

12 

35 

16 

35 

17 

13 

148 

352 

563 

155 

182 

202 

31 

211 

13 

38 

17 

38  ' 

18 

14 

160 

379 

607 

167 

196 

217 

34 

228 

14 

41 

18 

41 

20 

15 

171 

406 

650 

179 

210 

233 

36 

244 

15 

44 

19 

44 

21 

16 

182 

433 

693 

191 

224 

248 

38 

260 

16 

47 

21 

47 

23 

17 

194 

460 

737 

203 

238 

264 

41 

276 

17 

50 

22 

50 

24 

18 

205 

487 

780 

215 

252 

279 

43 

293 

18 

53 

23 

53 

25 

19 

217 

515 

823 

227 

266 

295 

46 

309 

19 

56 

25 

56 

27 

20 

228 

542 

867 

239 

280 

310 

48 

325 

20 

58 

26 

53 

28 

21 

239 

569 

910 

251 

294 

326 

50 

341 

21 

61 

27 

61 

30 

22 

251 

596 

953 

263 

308 

341 

53 

358 

22 

64 

28 

64 

31 

23 

262 

623 

997 

275 

322 

357 

55 

374 

23 

67 

30 

67 

33 

24 

274 

650 

1040 

287 

336 

372 

58 

390 

24 

70 

31 

70 

34 

Hours. 

Evection. 

Anomaly. 

Variation. 

Longitude. 

0    '    " 

0    '    // 

O    '    " 

0    '    "  * 

1 

0  28  17 

0  32  39.7 

0  30  29 

0  32  56.5 

2 

0  56  35 

1  5  19.5 

1  0  57 

1  5  52.9 

3 

1  24  52 

1  37  59.2 

1  31  26 

1  38  49.4 

4 

1  53  10 

2  10  39.0 

2  1  54 

2  11  45.8 

5 

2  21  27 

2  43  18.7 

2  32  23 

2  44  42.3 

6 

2  49  45 

3  15  58.5 

3  2  52 

3  17  38.8 

7 

3  18  2 

3  48  38.2 

3  33  20 

3  50  35.2 

8 

3  46  20 

4  21  18.0 

4  3  49 

4  23  31.7 

9 

4  14  37 

4  53  57.7 

4  34  17 

4  56  28.1 

10 

4  42  55 

5  26  37.5 

5  4  46 

5  29  24.6 

11 

5  11  12 

5  59  17.2 

5  35  15 

6  2  21.0 

12 

5  39  30 

6  31  57.0 

6  5  43 

6  35  17.5 

13 

6  7  47 

7  4  36.7 

6  36  12 

7  8  14.0 

14 

6  36  5 

7  37  16.5 

7  6  40 

7  41  10.4 

15 

7  4  22 

8  9  56.2 

7  37  9 

8  14  6.9 

16 

7  32  40 

8  42  36.0 

8  7  38 

8  47  3.4 

17 

8  0  57 

9  15  15.7 

8  38  6 

9  19  59.8 

18 

8  29  15 

9  47  55.5 

9  8  35 

9  52  56.3 

19 

8  57  32 

10  20  35.2 

9  39  3 

10  25  52.7 

20 

9  25  50 

10  53  15.0 

10  9  32 

10  58  49.2 

21 

9  54  7 

11  25  54.7 

10  40  1 

11  31  45.6 

22 

10  22  24 

11  58  34.5 

11  10  29 

12  4  42.1 

23 

10  50  42 

12  31  14.2 

11  40  58 

12  37  38.6 

24 

11  18  59 

13  3  54.0 

12  11  27 

13  10  35.0 

TABLE.   XXXVIII. 


59 


Moon's  Motions  for  Hours. 


H. 

14 

15 

16 

17 

18  ||  19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

1 

4 

1 

2 

2 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

2 

8 

3 

3 

4 

2 

0 

0 

0 

0 

1 

0 

0 

0 

1 

0 

0 

3 

12 

4 

5 

5 

3 

0 

0 

0 

0 

1 

0 

1 

0 

1 

0 

4 

16 

5 

6 

7 

4 

0 

0 

0 

0 

2 

0 

1 

1 

1 

1 

5 

21 

6 

8 

9 

5 

0 

0 

0 

0 

2 

1 

1 

1 

1 

6 

25 

8 

9 

11 

6 

0 

0 

0 

0 

3 

1 

1 

1 

2 

7 

29 

9 

11 

12 

8 

1 

0 

0 

0 

3 

1 

1 

1 

2 

8 

33 

10 

12 

14 

9 

1 

0 

0 

0 

3 

1 

1 

| 

2 

9 

37 

11 

14 

16 

10 

j  1 

0 

0 

0 

4 

1 

2 

1 

3 

1 

10 

41 

13 

15 

18 

11 

1 

0 

0 

0 

4 

i 

2 

2 

3 

2 

11 

45 

14 

17 

19 

12 

1 

0 

0 

0 

5 

i 

2 

2 

3 

2 

12 

49 

15 

18 

21 

13 

1 

0 

0 

0 

5 

i 

2 

2 

3 

2 

2 

13 

54 

16 

20 

23 

14 

1 

0 

0 

0 

5 

i 

2 

2 

4 

2 

2 

14 

58 

18 

21 

25 

15 

1 

0 

0 

0 

6 

2 

2 

2 

4 

2 

2 

15 

62 

19 

23 

26 

16 

1 

0 

0 

0 

6 

2 

3 

2 

4 

3 

2 

16 

66 

20 

25 

28 

17 

1 

0 

1 

1 

7 

2 

3 

2 

5 

3 

2 

17 

70 

21 

26 

30 

18 

1 

0 

1 

7 

2 

3 

1 

5 

3 

2 

18 

74 

23 

28   32 

19 

2 

0 

1 

8 

2 

3 

3 

5 

3 

2 

19 

78 

24 

29   33 

21 

2 

0 

1 

8 

2 

3 

3 

6 

3 

3 

20 

83 

25 

31   35 

22 

2 

0 

1 

8 

2 

3 

3 

6 

3 

3 

21 

87 

26 

32   37 

23 

2 

0 

1 

9 

2 

4 

3 

6 

4 

3 

22 

91 

28 

34   39 

24 

2 

0 

1 

9 

2 

4 

3 

6 

4 

3 

23 

95   29 

35 

40 

25 

2 

0 

1 

10 

3 

4 

4 

7 

4 

3 

24 

99 

31 

37 

42 

26 

2   0 

1 

10 

3 

4 

4 

7 

4 

3 

H. 

Sap.  of  Nod. 

II 

V 

VI 

VII 

VIII 

IX 

X 

XI 

XII 

,   „ 

0    ' 

1 

0   7.9 

0  28 

1 

2 

1 

1 

1 

0 

2* 

1 

2 

0  15.9 

0  56 

3 

3 

2 

3 

3 

0 

? 

3 

3 

0  23.8 

1  24 

4 

5 

4 

4 

4 

1 

5 

4 

4 

0  31.8 

1  52 

6 

7 

5 

6 

6 

1 

7 

6 

5 

0  39.7 

2  19 

7 

8 

6 

7 

7 

1 

9 

7 

6 

0  47.7 

2  47 

9 

10 

7 

9 

9 

1 

10 

9 

7 

0  55.6 

3  15 

10 

12 

8 

10 

10 

2 

12 

10 

8 

1   3.6 

3  43 

11 

13 

9 

11 

12 

2 

14 

11 

9 

11.5 

4  11 

13 

15 

11 

13 

13 

2 

15 

13 

10 

19.4 

4  39 

14 

16 

12 

14 

15 

2 

17 

14 

11 

27.4 

5  7 

16 

18 

13 

15 

16 

2 

19 

15 

12 

35.3 

5  35 

17 

20 

14 

17 

18 

3 

21 

17 

13 

43.3 

6  2 

18 

21 

15 

18 

19 

3 

23 

18 

14 

51.2 

6  30 

20 

23 

16 

19 

21 

3 

24 

19 

15 

59.2 

6  58 

21 

25 

18 

21 

22 

3 

26 

21 

16 

2  7.1 

7  26 

23 

26 

19 

22 

24 

4 

28 

22 

17 

2  15.0 

7  54 

24 

28 

20 

24 

25 

4 

29 

24 

18 

2  23.0 

8  22 

26 

29 

21 

25 

27 

4 

31 

25 

19 

2  30.9 

8  50 

27 

31 

22 

27 

28 

4 

33 

27 

20 

2  38.9 

9  18 

28 

32 

24 

28 

30 

4 

35 

28 

21 

2  46.8 

9  45 

30 

34 

25 

29 

31 

5 

37 

29 

22 

2  54.8 

10  13 

31 

36 

26 

31 

33 

5 

38 

31 

23 

3  2.7 

10  41 

33 

38 

27 

32 

34 

5 

40 

32 

24 

3  10.6 

11   9 

34 

39 

28  1 

34 

36 

5 

42 

34 

TABLE  XXXIX. 


Moon's  Motions  for  Minutes. 


1 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

1 

0 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

2 

0 

I 

1 

0 

0 

1 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

3 

1 

1 

2 

1 

1 

1 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

4 

1 

2 

3 

1 

1 

1 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

5 

1 

2 

4 

1 

1. 

1 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

6 

1 

3 

4 

1 

1 

2 

0 

2 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

7 

1 

3 

5 

1 

2 

2 

0 

2 

0 

0  . 

0 

0 

0 

0 

0 

0 

0 

0 

8 

2 

4 

6 

2 

2 

2 

0 

2 

0 

0 

0 

0 

0 

0 

0 

0 

0 

9 

2 

4 

6 

2 

2 

2 

0 

2 

0 

0 

0 

0 

0 

0 

0 

0 

0 

10 

2 

5 

7 

2 

2 

3 

0 

3 

0 

0 

0 

0 

0 

0 

0 

0 

0 

11 

2 

5 

8 

2 

3 

3 

0 

3 

0 

0 

0 

0 

0 

0 

0 

12 

2 

5 

9 

2 

3 

3 

0 

3 

0 

0 

0 

0 

0 

0 

0 

13 

2 

6 

9 

3 

3 

3 

1 

4 

0 

0 

0 

0 

0 

0 

0 

14 

3 

6 

10 

3 

3 

4 

1 

4 

0 

0 

0 

0 

0 

0 

0 

15 

3 

7 

11 

3 

3 

4 

1 

4 

0 

0 

0 

0 

0 

0 

0 

16 

3 

7 

12 

3 

4 

4 

1 

4 

0 

0 

0 

0 

0 

0 

0 

17 

3 

8 

12 

3 

4 

4 

1 

5 

0 

0 

0 

0 

0 

0 

0 

18 

3 

8 

13 

4 

4 

5 

1 

5 

0 

0 

0 

0 

0 

1 

0 

19 

4 

9 

14 

4 

4 

5 

1 

5 

0 

0 

0 

0 

0 

1 

0 

20 

4 

9 

14 

4 

5 

5 

1 

5 

0 

0 

0 

0 

1 

1 

0 

21 

4 

10 

15 

4 

5 

5 

1 

6 

0 

0 

0 

1 

0 

1 

1 

0 

22 

4 

10 

16 

4 

5 

6 

1 

6 

0 

0 

1 

2 

0 

1 

1 

0 

23 

4 

10 

17 

5 

5 

6 

1 

6 

0 

0 

1 

2 

0 

1 

1 

0 

24 

5 

11 

17 

5 

6 

6 

1 

7 

0 

1 

2 

1 

1 

1 

0 

»25 

5 

11 

18 

5 

6 

6 

1 

7 

0 

1 

2 

1 

1 

1 

0 

26 

5 

12 

19 

5 

6 

7 

1 

7 

0 

1 

2 

1 

1 

1 

0 

27 

5 

12 

19 

5 

6 

7 

1 

7 

0 

1 

2 

1 

1 

1 

0 

28 

5 

13 

20 

6 

7 

7 

1 

8 

0 

1 

2 

1 

1 

1 

0 

29 

6 

13 

21 

6 

7 

7 

1 

8 

0 

1 

1 

I 

2 

1 

1 

1 

0 

,30 

6 

14 

22 

6 

7 

8 

1 

8 

0 

1 

1 

1 

2 

1 

1 

1 

0 

TABLE  XXXIX. 


61 


Moon's  Motions  for  Minutes. 


i 

Sup. 

Min. 

Evec. 

Anom. 

Varia. 

Long. 

Nod. 

II 

V 

VI 

VII 

vm 

IX 

XI 

XII 

1 

0  28 

0  32.7 

0  30 

0  32.9 

0.1 

0 

0 

0 

0 

0 

0 

0 

0 

2 

0  57 

1  5.3 

1  1 

1  5.9 

0.3 

1 

0 

0 

0 

0 

0 

0 

0 

3 

1  25 

1  38.0 

1  31 

1  38.8 

0.4 

1 

0 

0 

0 

0 

0 

0 

0 

4 

1  53 

2  10.6 

2  2 

2  11.8 

0.5 

2 

0 

0 

0 

0 

0 

0 

0 

5 

2  2] 

2  43.3 

2  32 

2  44.7 

0.7 

2 

0 

0 

0 

0 

0 

0 

0 

6 

2  50 

3  16.0 

3  3 

3  17.6 

0.8 

3 

0 

0 

0 

0 

0 

0 

0 

7 

3  18 

3  48.6 

3  33 

3  50.6 

0.9 

3 

0 

0 

0 

0 

0 

0 

0 

8 

3  46 

4  21.3 

4  4 

4  23.5 

1.1 

4 

0 

0 

0 

0 

0 

0 

0 

9 

4  15 

4  54.0 

4  34 

4  56.5 

1.2 

4 

0 

0 

0 

0 

0 

0 

0 

10 

4  43 

5  26.6 

5  5 

5  29.4 

1.3 

5 

0 

0 

0 

0 

0 

0 

0 

11 

5  11 

5  59.3 

5  35 

6  2.4 

1.5 

5 

0 

0 

0 

0 

0 

0 

0 

12 

5  40 

6  31.9 

6  6 

6  35.3 

1.6 

6 

0 

0 

0 

0 

0 

0 

0 

13 

6  8 

7  4.6 

6  36 

7  8.2 

1.7 

6 

0 

0 

0 

0 

0 

0 

0 

14 

6  36 

7  37.3 

7  7 

7  41.2 

1.9 

7 

0 

0 

0 

0 

0 

0 

0 

15 

7  4 

8  9.9 

7  37 

8  14.1 

2.0 

7 

0 

0 

0 

0 

0 

0 

0 

16 

7  33 

8  42.6 

8  8 

8  47.1 

2.1 

7 

0 

0 

0 

0 

0 

0 

0 

17 

8  1 

9  15.3 

8  38 

9  20.0 

2.3 

8 

0 

0 

0 

0 

0 

0 

0 

18 

8  29 

9  47.9 

9  9 

9  52.9 

2.4 

8 

0 

0 

0 

0 

0 

1 

0 

19 

8  58 

10  20.6 

9  39 

10  25.9 

2.5 

9 

0 

0 

0 

0 

0 

1 

0 

20 

9  26 

10  53.2 

10  10 

10  58.8 

2.6 

9 

0 

1 

0 

0 

0 

1 

0 

21 

9  54 

11  25.9 

10  40 

11  31.8 

2.8 

10 

0 

0 

0 

0 

1 

0 

22 

10  22 

11  58.6 

11  11 

12  4.7 

2.9 

10 

1 

0 

0 

1 

1 

0 

23 

10  51 

12  31.2 

11  41 

12  37.6 

3.0 

11 

1 

0 

0 

1 

1 

0 

24 

11  19 

13  3.9 

12  12 

13  10.6 

3.2 

11 

1 

0 

1 

1 

1 

25 

11  47 

13  36.6 

12  42 

13  43.5 

3.3 

12 

1 

0 

1 

1 

1 

26 

12  16 

14  9.2 

13  13 

14  16.5 

3.4 

12 

1 

1 

1 

27 

12  44 

14  41.9 

13  43 

14  49.4 

3.6 

13 

1 

1 

1 

28 

13  12 

15  14.6 

14  13 

15  22.3 

3.7 

13 

1 

1 

1 

29 

13  40 

15  47.2 

14  44 

15  55.3 

3.8 

13 

1 

1 

1 

30 

14  9 

16  19.9 

15  14 

16  28.2 

4.0 

14 

1 

1 

1 

TABLE  XXXIX. 


Moon's  Motions  for  Minutes. 


1 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

! 
18 

31 

6 

14 

22 

6 

7 

8 

8 

0 

1 

1 

2 

1 

32 

6 

14 

23 

6 

7 

8 

9 

1 

2 

2 

2 

33 

6 

15 

24 

7 

8 

9 

i 

9 

1 

2 

2 

2 

34 

6 

15 

25 

7 

8 

9 

9 

1 

2 

2 

2 

35 

7 

1C 

25 

7 

8 

9 

10 

1 

2 

2 

2 

30 

7 

16 

26 

7 

8 

9 

1 

10 

1 

2 

2 

3 

37 

7 

17 

27 

7 

9 

10 

1 

10 

2 

2 

3 

38 

7 

17 

27 

8 

9 

10 

2 

10 

2 

2 

3 

39 

7 

18 

28 

8 

9 

10 

2 

11 

2 

2 

3 

40 

8 

18 

29 

8 

9 

10 

2 

11 

2 

2 

3 

41 

8 

19 

30 

8 

10 

11 

2 

11 

2 

2 

3 

42 

8 

19 

30 

8 

10 

11 

2 

11 

2 

2 

3 

43 

8 

19 

31 

9 

10 

11 

o 

12 

2 

2 

3 

44 

8 

20 

32 

9 

10 

11 

2 

12 

2 

2 

3 

45 

9 

20 

32 

9 

10 

12 

2 

12 

2 

2 

3 

46 

9 

21 

33 

9 

11 

12 

2 

12 

2 

2 

3 

47 

9 

21 

34 

9 

11 

12 

2 

13 

2 

2 

3 

48 

9 

22 

35 

10 

11 

12 

2 

13 

2 

2 

3 

49 

9 

22 

35 

10 

11 

13 

2 

13 

2 

2 

3 

50 

9 

23 

36 

10 

11 

13 

2 

13 

2 

2 

3 

51 

10 

23 

37 

10 

12 

13 

2 

14 

2 

2 

4 

52 

10 

24 

38 

10 

12 

13 

2 

14 

3 

3 

4 

53 

10 

24 

38 

11 

12 

14 

2 

14 

3 

3 

4 

54 

10 

24 

39 

11 

12 

14 

2 

14 

3 

3 

4 

2 

55 

10 

25 

40 

11 

13 

14 

2 

15 

3 

3 

4 

1 

2 

56 

11 

25 

40 

11 

13 

14 

2 

15 

3 

3 

4 

1 

2 

57 

11 

26 

41 

11 

13 

15 

2 

15 

3 

3 

4 

1 

2 

58 

11 

26 

42 

12 

13 

15 

2 

16 

3 

3 

4 

2 

2 

59 

11 

27 

43 

12 

14 

15 

2 

16 

3 

3 

4 

2 

2 

j 

60 

11 

27 

43 

12 

14 

15 

2 

16 

1 

3 

3 

4  1 

2 

2 

TABLE  XXXIX. 
Moorfs  Motions  for  Minutes. 


63 


Sup. 

Min 

Evec. 

Anom. 

Varia. 

Long. 

Nod. 

II 

V 

VI 

VII 

vm 

IX 

XI 

XII 

31 

14  37 

16  52.5 

15  45 

17  1.2 

4.1 

14 

1 

1 

i 

1 

32 

15  5 

17  25.2 

16  15 

17  34.1 

4.2 

15 

1 

1 

i 

1 

33 

15  34 

17  57.9 

16  46 

18  7.1 

4.4 

15 

1 

1 

i 

1 

34 

16  2 

18  30.5 

17  16 

18  40.0 

4.5 

16 

1 

1 

i 

1 

35 

16  30 

19  3.2 

17  47 

19  12.9 

4.7 

16 

1 

1 

i 

1 

36 

16  58 

19  35.8 

18  17 

19  45.9 

4.8 

17 

1 

1 

i 

37 

17  27 

20  8.5 

18  48 

20  18.8 

4.9 

17 

1 

1 

i 

38 

17  55 

20  41.2 

19  18 

20  51.8 

5.0 

18 

1 

1 

i 

39 

18  23 

21  13.8 

19  49 

21  24.7 

5.2 

18 

1 

1 

i 

40 

18  52 

21  46.5 

20  19 

21  57.6 

5.3 

19 

1 

1 

i 

41 

19  20 

22  19.2 

20  50 

22  30.6 

5.4 

19 

1 

i 

42 

19  48 

22  51.8 

21  20 

23  3.5 

5.6 

20 

1 

i 

43 

20  16 

23  24.5 

21  51 

23  36.5 

5.7 

20 

1 

i 

44 

20  45 

23  57.1 

22  21 

24  9.4 

5.8 

21 

1 

i 

45 

21  13 

24  29.8 

22  52 

24  42.3 

6.0 

21 

1 

i 

i 

46 

21  41 

25  2.5 

23  22 

25  15.3 

6.1 

21 

1 

i 

47 

22  10 

25  35.1 

23  53 

25  48.2 

6.2 

22 

1 

i 

48 

22  38 

26  7.8 

24  23 

26  21.2 

6.4 

22 

1 

i 

49 

23  6 

26  40.5 

24  54 

26  54.1 

6.5 

23 

1 

i 

50 

23  34 

27  13.1: 

25  24 

27  27.0 

6.6 

23 

1 

i 

51 

24  3 

27  45.8 

25  55 

28  0.0 

6.8 

24 

1 

i 

52 

24  31 

28  18.5 

26  25 

28  32.9 

6.9 

24 

1 

i 

53 

24  59 

28  51.1 

26  56 

29  5.9 

7.0 

25 

1 

1 

i 

54 

25  28 

29  23.8 

27  26 

29  38.8 

7.1 

25 

1 

1 

i 

2 

55 

25  56 

29  56.4 

27  56 

30  11.8 

7.3 

26 

1 

1 

i 

2 

56 

2f>  24 

30  29.1 

28  27 

30  44.7 

7.4 

26 

1 

1 

i 

2 

57 

26  52 

31  1.8 

28  57 

31  17.6 

7.5 

27 

2 

1 

i 

2 

58 

27  21 

31  34.4 

29  28 

31  506 

7.7 

27 

2 

1 

i 

2 

59 

27  49 

32  7.1 

29  58 

32  &I..5 

7.8 

28 

2 

1 

i 

2 

60 

28  17 

32  39.8 

SO  29 

32  56.5 

7.9 

28 

2 

1 

i 

2 

1 

;  .1 
64 

I 


TABLE  XL. 


Moon's  Motions  for  Seconds. 


Sec. 

Evec. 

Anom. 

Var. 

Long. 

Sec. 

Evec. 

Anom. 

,Var. 

Long. 

1 

0 

0.5 

1 

0.5 

31 

15 

16.9 

16 

17.0 

2 

1 

1.1 

1 

1.1 

32 

15 

17.4 

16 

17.6 

3 

1 

1.6 

2 

1.6 

33 

16 

18.0 

17 

J8.1 

4 

2 

2.2 

2 

2.2 

34 

16 

18.5 

17 

18.7 

5 

2 

2.7 

3 

2.7 

35 

17 

19.1 

18 

19.2 

6 

3 

3.3 

3 

3.3 

36 

17 

19.6 

18 

19.8 

7 

3 

3.8 

4 

3.8 

37 

18 

20.1 

19 

20.3 

8 

4 

4.3 

4 

4.4 

38 

18 

20.7 

19 

20.9 

9 

4 

4.9 

5 

4.9 

39 

18 

21.2 

20 

21.4 

10 

5 

5.4 

5 

5.5 

40 

19 

21.8 

20 

22.0 

11 

5 

6.0 

6 

6.0 

41 

19 

22.3 

21 

22.5 

12 

6 

6.5 

6 

6.6 

42 

20 

22.9 

21 

23.1 

13 

6 

7.1 

7 

7.1 

43 

20 

23.4 

22 

23.6 

14 

7 

7.6 

7 

7.7 

44 

21 

24.0 

22 

24.2 

15 

7 

8.2 

8 

8.2 

45 

21 

24.5 

23 

24.7 

16 

8 

8.7 

8 

8.8 

46 

22 

25.0 

23 

25.3 

17 

8 

9.2 

9 

9.3 

47 

22 

25.6 

24 

25.8 

18 

9 

9.8 

9 

9.9 

48 

23 

26.1 

24 

26.4 

19 

9 

10.3 

10 

10.4 

49 

23 

26.7 

25 

26.9 

20 

9 

10.9 

10 

11.0 

50 

24 

27.2 

25 

27.4 

21 

10 

11.4 

11 

11.5 

51 

24 

27.8 

26 

28.0 

22 

10 

12.0 

11 

12.1 

52 

25 

28.3 

26 

28.5 

23 

11 

12.5 

12 

12.6 

53 

25 

28.9 

27 

29.1 

24 

11 

13.1 

12 

13.2 

54 

26 

29.4 

27 

29.6 

25 

12 

13.6 

13 

13.7 

55 

26 

29.9 

28 

30.2 

26 

12 

14.1 

13 

14.3 

56 

26 

30.5 

28 

30.7 

27 

13 

14.7 

14 

14.8 

57 

27 

31.0 

29 

31.3 

28 

13 

15.2 

14 

154 

58 

27 

31.6 

29 

31.8 

29 

14 

15.8 

15 

15.9 

5" 

28 

32.1 

30 

32.4 

SO 

14 

16.3 

15 

16.5 

60 

28 

32.7 

30 

32.9  j 

TABLE  XLI,  65 

First  Equation  of  Moon's  Longitude*^ Argument  1. 


Diff. 

Diff.  • 

Diff. 

Diff. 

Arg. 

1 

for  10 

Ar& 

1 

for  10 

Arg, 

1 

for  10 

Arg. 

1 

for  10 

0 
50 
100 
150 
200 
850 

12  40.0 
12  18.8 
11  57.7 
11  36.fi 
11  15.6 
10  &4.7 

4.24 

432 
4.S2 
480 
4.1.8 
4.16 

8500 
2550 
SGOO 
2050 
2700 
2750 

1  40.7 
1  41.5 

1  42.9 
1  45.0 
1  47.7 
1  51.C 

0  16 

0.28 
0.42 
0.54 
0.66 
0.80 

5000 
5050 
5100 
r>K>0 
6200 
5260 

12  40.0 
13  0.3 
13  20.6 
13  40.7 
14  0.0 
14  20.9 

4.06 
4.04 
4.04 
404 
4.00 
4.00 

750023  39.3 
7550  i  23  39.4 
7BOO'23  38.9 
765023  37.7 
770023  35.8 
7750*23  33.3 

0.02 

0.24 
0.38 
0.50 
0.62 

300 
350 
400 
450 
500 

10  33.9 
10  13.2 
9  55.6 
9  32.3 
9  1S.1 

4.14 
413 
4.06 
4.04 
4,00 

2SOO 
2850 
2900 
^950 
3000 

1  55.0 
1  59.6 

U  4.8 
2  10.7 
2  17.1 

0.92 
1.04 

1.28 
1.42 

^300 
5350 

5450 
5500 

14  40.9 
15  0.8 
15  20.5 
15  40.1 
15  59.6 

3.98 
3.9-4 
3.92 
3.90 
B,84 

780023  30.2 
785023  26.4 
790023  22.0 
795023  16.9 
8000p  11.2 

O.V6 
0.88 
1.02 
1.14 
1.26 

550 
600 
650 

8  52.1 
8  32.4 
8  13.0 

3.94 

3.88 

3050 
5100 

2  24.2 
2  3L9 
2  40.1 

i.M 

1.64 

5550 
5600 

16  18.8 
16  56.7 

3.80 
3,78 

8050  23  4.9 
810022  57.9 
815022  50.3 

1.40 
1.52 

700 
750 

7  53.8 
V  34.9 

3.84 
3.78 
3.70 

3200 

2  48.9 
2  58.8 

1.76 
L.88 
1,99 

5700 

17  15.3 
17  33.6 

3.72 
3.66 
3.60 

8200 

8250 

22  42.0 
22  33.2 

1.66 
1.76 
1.9.0 

800 
850 

t  16.4 
6  58.2 

3.64 

3300 
1350 

3  8.2 
3  18.7 

2,10 

5800 
5850 

17  51.6 
18  9.4 

3.56 

830022  23.7 
835022  13.7 

2-.00 

900 
950 
1000 

6  40.3 
6  22.8 
6  5.7 

3.60 
3.42 
3.34 

3400 
3450 
3-500 

3  4L3 
3  53.4 

2.32 
2.42 
2.50 

5900 
5950 
6000 

18  26.9 
18  44.0 
19  0.8 

3.50 
3.42 
3.36 
3.28 

840022  3.1 
845021  51.9 
8500  2-1-  40.1, 

2.-  12 
2.24 
2.36 
246 

1050 

•150 
,200 
1250 

5  49.0 
5  32.8 
5  17.0 
5  1.6 
4  46.7 

3.34 
3,16 
3.08 

2.98 
2.88 

3550 
3600 
3650 
3700 
3750 

4  5.9 
4  19.0 
4  32.5 
4  46.5 
5  0.9 

2.62 
2.70 

2.80 
2.88 
2.98 

6050 
6100 
6150 
6200 
6250 

19  17.2 
19  33.3 
19  49.£ 
20  4.2 
80*  19.1 

3.22 
3.14- 
3.04 
2.98 

2.88 

8950 
8600 
S650 
1  8700 
8750 

2J  278 
21  15.0 
21  1.6 
20  47.7 
20  33.3 

2.56 

2,68 
2-.  78 
2.88 
2.98 

1300 
I3f)0 
1400 
1450 
1500 

4  32.3 
4  18.4 
4  5.0 
3  52.2 
3  39.9 

2.78 
2.68 
2.56 
2.46 
2.36 

3800 
3850 
3900 
3950 
4000 

5  15.8 
5  31.0 
5  46.7 
6  2.8 
6  1-9.  » 

3.04 

3.14- 
3.22- 
3.38 
3.36 

6800 
6350 
6-400 
6450 
6500 

2Q  33.5 
20  47.5 
21  1.0 
21  14.1 
21  26.6 

2.80 
2.70 
2.62 
2.50 
2.42 

880020  18.4 
8850  20  3.0 
8900  19  47.2 
8950  19  31.0 
9000  19  143 

3.08 
3.16 
3.24 
3.34 
3.42 

1550 

3  28.1 

2  24 

4050 

6-  36.0 

o  40 

6550 

21  38.7 

sy  qo 

9050  18  57.2 

3KA 

1600 
1650 
1700 
1750 

3  16.9 
3  6.3 
2  56.3 
2  46.8 

2.12. 
2.00 
1.90 
1.78 

4150 
4200 
4250 

6  53.1 
7  10.6- 

7  28.4 
7  46.4 

350 
3.56 
3.60 
3.66 

6600 
6650 
16700 
0750 

21  50.3 
22  1.3 

22  11.8 
22  21.7 

2.20 
2.10 
1.98 
1.88 

9100  18  39.7 
9150  18  21.8 
9200  18  8.6 
9250  IT  43:  1 

3.58 
3.84 
3.70 
3.78 

1800 

2  38.0 

1  fift 

430d  8  4.7 

6800 

22  31.1 

7ft 

9300 

17  26.2 

q  QX 

1850 
1900 
1950 
2000 

2  29.7 
2  22.1 
2  15.1 
2  8.8 

1.52: 
1.40 
1.26 
1.14 

4350 
4400 
4450 
4500 

8  23.3 

8  42.2 
9  1.2 
9  20.4 

3.78 
3.80 
3.84 
3.90 

685C 
690C 
G95C 
70QC 

22  39.9 

22  48.1 
22  55.8 

23  2.9 

1 

.64 
.54 

.42 
.28 

9350 
9400 
9450 
9500 

17  7,0 
16  47.6 
16  27.9 
16  7:9 

'3.88 
3.94 
4.00 
4.04 

2050 
2100 
2130 
2200 
2230 

2  3.1 
58.0 
53.6 
49.8 
46.7 

1.02 
0.88 
0.76 
0.62 
0.50 

4550 
4600 
4650 
4700 
475C 

9  39.9 
9  59.5 
10  19.2 
10  39.1 
10  59.1 

3.92 
3.94 
3.98 
4.00 
4.00 

705023  9.3 
7100J23  15.2 
7130  23-  2#.4 
720023  25.0 
725023  29.0 

.18 
.04 
0.92 
0.80 
0.66 

9550 
9600 
9650 
9700 
9750 

15  47:7 
15,27.4 
15  6.8 
14  46.1 
14  25:3 

406 
4.12 
4,14 
416 
4.18 

2300    44.2 
2350!   42.3 
24001   41.1 
2450    40  fi 

0.38 
0.24 
0.10 

480C 
485C 
490C 
495(] 

11  19.1 
11  39.3 
11  59.S 
12  19  1 

4.04 
4.04 
4.04 

7300  23  32.3 
7350,23  35.0 
740023  37.1 
7450  23  38  5 

0.54 
0.42 

0.28 

9800 
9850 
9900 
9950 

14  4;4 
13  43-4 
|13.  2-2-.  3 

1A-  1  9 

4.22-; 

4.22: 

2500    40  7  0.02 

5000  IS  40.  ft  ^U  U 

750023  39.3  °'lb 

1.0000  12  40,0 

4.1& 

66  TABLE  XLII. 

Equations  2  to  7  of  Moon's  Longitude.     Arguments  2  to  7 


Arg. 

2 

diff 

3 

diff  I     4 

diff 

5 

diff 

6 

diff 

7 

diff 

1 
Arg. 

/    ,f 

/    // 

/    „ 

,    // 

/    // 

/    „ 

2500 
2600 
2700 

457.3 

457.0 
456.1 

0.3 

0.9 

0   2.3 

0   2.4 
0    2.8 

0.1 

0.4 

630.3 
629.9 
628.8 

0.4 
1.1 

339.4 
339.2 
338.5 

0.2 
0.7 

0   6.2 

* 

0   0.8 
0   0.9 
0    1.3 

0.1 

0.4 

2500 
2400 
2300 

2800 

454.7 

1.4 

0    3.3 

0.5 

A   O 

626.9 

1.9 

3  37.5 

1.0 

j     ^lU.O 

0    1.8!™  2200 

2900 

4  52.7 

2.0 

O   ft 

0   4.1 

0.8 

i  n 

624.3 

2.6 

q   0 

336.0 

1.5 

1    o 

0    8.8  ,'J 

0    2.7 

?•;  2100 

3000 

450.1 

2'6  0    5.1 

l.U 

621.0 

o.o 

334,1 

i.y 

010.3 

0    3.7 

IM  2000 

3.1 

1.3 

4.1 

2.4 

1.8 

1.3 

3100 

447.0 

3  7 

0    6.4 

J.4 

616.9 

4,7 

331.7 

2  7 

012.1 

2.1 

0    5.0 

1.4 

1900 

3200 
3300 
3400 
3500 

443.3 
439.1 
434.4 

4292 

4.2 
4.7 
5.2 

0    7.8 
0    9.4 
011.3 
013.3 

1.6 
1.9 
2.0 

612.2 
6    6.8 
6    0.7 
5  54,0 

5.4 
6.1 
6.7 

3  29.0 
3  25.9 
322.4 
318.5 

31 
3.'5 
3.9 

014.2 
016.6 
019.2 
022.2 

2^4 
2.6 
3.0 

0    6.4 
0   8.1 
010.0 
012.1 

1.7 
1.9 
2.1 

1800 
1700 
1600 
1500 

5.7 

2.2 

7.4 

4.2 

3.2 

2.3 

3600 

4235 

6.1 

0  15.5 

2.4 

546.6 

7.9 

314.3 

4.6 

025.4 

3.5 

014.4 

2.4 

1400 

3700 
3800 

417.4 
410.8 

6.6 
6.9 

017.9 
020.5 

2.6 

2.7 

538.7 
530.3 

8.4 
9.0 

3    9.7 
3    4.9 

4.8 
5.2 

028.9 
032.7 

3^8 
3.9 

016.8 
0  19.5 

2.7 
2.8 

1300 
1200 

3900 

4    3.9 

7.3 

023.2 

521.3 

Q  4. 

259.7 

5  4 

0  36.6 

A.  * 

022.3 

2.9 

1100 

4000 

356.6 

7.7 

026.1 

3.0 

511.9 

y.*r 
9.9 

2543 

5.7 

°40.7- 

025.2 

3.1 

1000 

4100 

348.9 

7.9 

029.1 

3.1 

5    2.0 

10.3 

24S.P 

5.9 

045.1 

4.5 

028.3 

3.2 

900 

4200 

341.0 

8.3 

032.2 

3.2 

451.7 

10  7 

242.7 

6.1 

049.6 

4.7 

031  .6 

3.3 

800 

4300 

3  32.7 

8.5 

035.4 

3.4 

441.0 

10.9 

236.6 

6.3 

0  54.3 

4.9 

034.8 

3.4 

700 

4400 

324.2 

8.7 

038.8 

3.4 

430.1 

Uo 

230.3 

6.5 

0  £9.2 

4.9 

038.2 

3.5 

600 

4500 

3  15.5 

042.2 

418.8 

.  .0 

223.8 

1    4.1 

041.7 

500 

8.9 

3.5 

11.5 

6.6 

5.1 

3.6 

I 

4600 

3    6.6 

9.0 

045.7 

3.5 

4    7.3 

11.6 

217.2 

6.7 

1    9.2 

5.1 

0  45.3 

3.6 

400! 

4700 

257.6 

9.1 

049.2 

3.6 

3  55.7 

11.8 

2  10.5 

3.8 

1  143 

5.2 

048.9 

3.7 

300j 

4800 
4900 

248.5 
239.2 

9.3 
9.2 

052.8 
0  56.4 

3.6 

343.9 
331.9 

12.0 
11.9 

2    3.7 
156.9 

6.8 
6.9 

1  19.5 
124.7 

5.2 
5.3 

052.6 
056.3 

3.7 
3.7 

100! 

5000 

230.0 

9.2 

1    0.0 

3.6 

320.0 

11.9 

150.0 

6.9 

130.0 

5.3 

1    0.0 

3.7 

0 

5100 

220.8 

93 

1    3.6 

3.6 

3    8.1 

12.0 

43.1 

6.8 

135.3 

5.2 

1    3.7 

3.7 

9900 

5200 

211.5 

9.1 

1    7.2 

3.6 

256.1 

11.8 

36.3 

6.8 

1  40.5 

5.2 

1    7.4 

3.7 

9800 

5300 

2    2.4 

9.0 

1  10.8 

3.5 

£44.3 

11.6 

29.5 

6.7 

1  45.7 

5.1 

1  11.1 

3.6 

9700 

15400 
!o500 

L53.4 

1  44.5 

8.9 

8.7 

1  14.3 
I  17.8 

3.5 
3.4 

232.7 

221.2 

11.0 
11.3 

22.8 
16.2 

6.6 
6.5 

150.8 
155.9 

5.1 
4.9 

1  14  7 
1  183 

3.6 
3.5 

9600 
9500 

5600 
5700 
5800 
5900 
{6000 

135.8 
127.3 
1  19.0 
1  11.1 
1    3.4 

S.5 
8.3 
7.9 
7.7 
7.3 

21.2 
124.6 
127.8 
130.9 
133.9 

3.4 
3.2 
3.1 

3.0 
2.9 

2    9.9 
1  59.0 
148.3 
1  38.0 
1  28.1 

10,9 
10.7 
10.3 
9.9 
9.4 

0.7 
3.4 
0  57.3 

045.7 

6.3 
6.1 
5.9 
5.7 
,5.4 

2    08 
2    5.7 
,210.4 
2149 
2193 

4.9 
4.7 
4,5 
4.4 
4.1 

151.8 
125.2 
128.5 
131.7 

1  34.8 

3.4 
3.3 
3.2 
3.1 
2.9 

9400J 
9300 
9200 
9100 
9000 

6100 
.6200 
j6300 
6400 
6500 

056.1 
049.2 
042.6 
0  36.5 
030.8 

6.9 
6.6 
6.1 
5.7 

136.8 
139.5 
142.1 
144.5 
146.7 

2.7 
2.6 
14 

2.2 

1  18.7 
1    9.7 
1    1.3 
0  53.4 
0  46.0 

9.0 

8.4 
7.9 
7.4 

0  40  3; 
035.1 
030.3 
025.7 
021.5 

5.2  3f'4 

1'fl  23L1 
Jo  «  34-6 

on  1337.8 

3.9 
3,8 
3.5 
3.2 

1  37.7 
1  40.5 
143.2 
145.6 

147.9 

B'JUU 
*-°  8800 

o  I  I8700 

2-4  86oo 
23  8500 

5.2 

2.0 

6.7 

3.9 

3.0 

2.1 

6600  0  25.6 
6700  0  20.9 
6800  0  16.7 
6900  0  13.0 
70000   9.9 

47 

4.2 
37 
3.1 

148.7 
1  50.6 
162.2 
153.6 
1549 

1.9 
1.6 
1.4 
1.3 

039.3 
033.2 
027.8 
023.1 
019.0 

6.1 
5.4 
4.7 
4.1 

017.6 
014.1 
011.0 
0    8.3 
0    5.9 

3.5 
3.1 
2.7 
2.4 

240.8 
243.4 
v>  45.8 
247.9 
249.7 

2.6 
2.4 

2.1 
1.8 

150.0 
I  51.9 
153.6 
155.0 
1563 

1.9 
1.7 
1.4 
1.3 

8400 
8300 
8200 
8100 
8000 

2.6 

1.0 

3.3 

1.9 

1.5 

1.0 

71000    7.3 
72000    5.3 
7300'.0    3.9 
74000   3.0 
75000    2.7 

2.0 
1.4 
0.9 
0.3 

155.9 

1  jB7,2 
157.6 
1  &7.7 

0.8 
05 

0.4 
0.1 

015.7 
013.1 
011.2 
010.1 
0   9.7 

2.6 
1.9 
1.1 
0.4 

0   4.0 
0    2.5 
0    1.5 
0   0.8 
0   0.6 

1.5 
1.0 
0.7 
0.2 

251.2 
2  52.3 
253.1 
2  53.6 
253.8 

1.1 
0.8 
0.5 
0.2 

157.3 
1  58.2 
158.7 
159.1 
159.2 

0.0 
0.5 
0.4 
0.1 

7900 
7800 
7700 
7600 
7500 

TABLE  XLIIL 


TABLE  XLIV.        C>7 


Equations  8  and  9, 


Equations  10  and  1J. 


Arg. 

8 

9 

Arg, 

8 

9 

0 

1  20.0 

1  1200 

5000 

1  20.0 

1  20.0 

100 

1  15.5 

1  287 

5100 

1  24.4 

1  25.8 

200 

1  11.1 

1  37.3 

5200 

1  28.8 

1  31.4 

300 

1  6.7 

1  45.7 

5300 

1  33.1 

1  36.9 

400 

1  2.3 

1  53.7 

5400 

1  37.4 

I  42.0 

500 

0  58.0 

2  1.3 

5500 

1  41.6 

1  46.8 

600 

0  53.8 

2  8.3 

5600 

1  45.8 

1  51.0 

700 

0  49.7 

2  14.7 

5700 

1  49.8 

1  54.6 

800 

0  45.7 

2  20.2 

5800 

1  53.8 

1  57.6 

900 

0  41.9 

2  25.0 

5900 

1  57.6 

1  59.8 

1000 

0  38.2 

2  28.9 

6000 

2  1.2 

2  1.3 

1100 

0  34.7 

2  31.9 

6100 

2  4.7 

2  1.9 

1200 

0  31.4 

2  33.9 

6200 

2  8.0 

2  1.7 

1300 

0  28.2 

2  34.9 

6300 

2  11.2 

2  0.7 

1400 

0  25.3 

2  35.0 

6400 

2  14.1 

1  58.8 

1500 

0  22.6 

2  34  1 

6500 

2  16.8 

1  56.1 

1600 

0  20.1 

2  32  2 

6600 

2  19.3 

1  52.5 

1700 

0  17.9 

2  29.5 

6700 

2  21.6 

1  48.3 

1800 

0  15.9 

2  25.9 

6800 

2  23.7 

1  43.4 

1900 

0  14.2 

2  21.5 

6900 

2  25.4 

1  37.8 

2000 

0  12.7 

2  16.4 

7000 

2  27.0 

1  31.7 

2100 

0  11.5 

2  10.7 

7100 

2  28.2 

1  25.1 

2200 

0  10.5 

2  4.4 

7200 

2  29.2 

1  18.2 

2300 

0  9.9 

1  57.7 

7300 

2  30.0 

1  11.  1 

2400 

0  9.5 

1  50.7 

7400 

2  30.4 

1  3.8 

2500 

0  9.4 

1  43.5 

7500 

2  30.6 

0  56.5 

2600 

0  9.6 

1  36.2 

7600 

2  30.5 

0  49.3 

2700 

0  10.1 

1  28.9 

7700 

2  30.1 

0  42.3 

2800 

0  10.8 

1  21.8 

7800 

2  29.5 

0  35.6 

2900 

0  11.8 

1  14.9 

7900 

2  28.5 

0  29.3 

3000 

0  13.0 

1  8.3 

8000 

2  27.3 

0  23.6 

3100 

0  14.6 

1  2.2 

8100 

2  25.8 

0  18.5 

3200 

0  16.3 

0  56.6 

8200 

2  24.1 

0  14.1 

3300 

0  18  4 

0  51.7 

8300 

2  22.1 

0  10.5 

3400 

0  20.7 

0  47.5 

8400 

2  19.9 

0  7.8 

3500 

0  23.2 

0  43.9 

8500 

2  17.4 

0  5.9 

3600 

0  25.9 

0  41.2 

8600 

2  14.7 

0  5.0 

3700 

0  28.8 

0  39.3 

8700 

2  11.8 

0  5.1 

3800 

0  32.0 

0  38.3 

8800 

2  8.6 

0  6.1 

3900 

0  35.3 

0  38.1 

8900 

2  5.3 

0  8.1 

4000 

0  38.8 

0  38.7 

9000 

2  1.8 

0  11.1 

4100 

0  42.4 

0  40.2 

9100 

58.1 

0  15.0 

4200 

0  46.2 

0  42.4 

9200 

54.3 

0  19.8 

4300 

0  50.2 

0  45.4 

9300 

50.3 

0  25.3 

4400 

0  54.2 

0  49.0 

9400 

46.2 

0  31.7 

4500 

0  58.4 

0  53.2 

9500 

42.0 

0  38.7 

4600 

1  26 

0  58.0 

9600 

37.7 

0  46.3 

4700 

1  6.9 

1  3.1 

9700 

33.3 

0  54.3 

4800 

1  11.2 

1  8.6 

9800 

28.9 

1  2.7 

4900 

1  15.6 

1  14.2 

9900 

24.5 

1  11.3 

5000 

1  20.0 

1  20.0 

10000 

I  20.0 

1  20.0 

Arg. 

10 

11 

Arg. 

10 

11 

0 

10.0 

10.0 

500 

10.0 

10.0 

10 

9.3 

11.1  i 

510 

9.6 

10.8 

20 

8.6 

12.1 

520 

9.2 

11.5 

30 

8.0 

13.1 

530 

8.9 

12.3 

40 

7.4 

14.1  ! 

540 

8.5 

12.9 

50 

6.8 

15.0 

550 

8.2 

13.6 

60 

6.2 

15.8 

560 

7.9 

14.2 

70 

5.7 

16.6 

570 

7.7 

14.6 

80 

5.3 

17.3 

580 

7.5 

15.0 

90 

4.9 

17.9 

590 

7.4 

15.4 

100 

4.6 

18.3  ' 

600 

7.3 

15.6 

110 

4.3 

18.6 

610 

7.2 

15.7 

120 

4.1 

18.9 

620 

7.3 

15.7 

130 

4.0 

19.0 

630 

7.4 

15.6 

140 

4.0 

18.9 

640 

7.5 

15.4 

150 

4.0 

18.8 

650 

7.8 

15.1 

160 

4.2 

18.6 

660 

8.1 

14.7 

170 

4.4 

18.2 

670 

8.4 

14.2 

180 

46 

17.7 

680 

8.7 

13.5 

190 

4.9 

17.1 

690 

9.2 

12.8 

200 

5.3 

16.5 

700 

9.7 

12.1 

210 

5.7 

15.7 

710 

10.2 

11.3 

220 

6.2 

14.9 

720 

10.7 

10.4 

230 

6.7 

14.1 

730 

11.2 

9.5 

240 

7.2 

13.2 

740 

11.7 

8.6 

250 

7.7 

12.3 

750 

12.3 

7.7 

260 

8.3 

11.4 

760 

12.8 

6.8 

270 

8.8 

10.5 

770 

13.3 

5.9 

280 

9.3 

9.6 

780 

13.8 

5.1 

290 

9.8 

8.7 

790 

14.3 

4.3 

300 

10.3 

7.9 

800 

14.7 

3.5 

310 

10.8 

7.2 

810 

15.1 

2.9 

320 

11.3 

6.5 

920 

15.4 

2.3 

330 

11.6 

3.8 

830 

15.6 

1.8 

340 

11.9 

5.3 

840 

1&.8 

1.4 

350 

12.2 

4.9 

850 

16.0 

1.2 

360 

12.5 

4.6 

860 

16.0 

1.1 

370 

12.6 

4.4 

1  870 

16.0 

1.0 

380 

12.7 

4.3 

880 

15.9 

1.1 

390 

12.8 

4.3 

890 

15.7 

1.4 

400 

12.7 

4.4 

900 

15.4 

1.7 

410 

12.6 

4.6 

910 

15.1 

2.1 

420 

12.5 

5.0 

920 

14.7 

2.7 

430 

12.3 

5.4 

930 

14.3 

3.4 

440 

12.1 

5.8 

940 

13.8 

4.8 

450 

11.8 

6.4 

950 

13.2 

5.0 

460 

11.5 

7.1 

960 

12.6  5.9 

470 

11.1 

7.7  i  970 

12.0  6.9 

480 

10.8 

8.5  i  980 

11.4  7.9 

490 

10.4 

9.2  j  910 

10.7  8.9 

500 

10.0 

10.0  !10(  0 

100  10.0 

TABLE  XLY. 
Equations  12  to  19. 


TABLE  XLVI. 
Equation  20. 


Arg. 

12 

13 

14 

15 

16 

17 

18 

19 

Arg. 

250 

2.3 

1.6 

•  7.8 

0.0 

33.7 

3.4 

16.7 

0.4 

250 

260 

2.3 

1.6 

7.8 

0.0 

33.7 

3.4 

16.7 

0.4 

240 

270 

2.4 

1.7 

7.9 

0.1 

33,6 

3.5 

16.6 

0.4 

230 

280 

2.6 

1.9 

8.0 

0.2 

33.5 

3.  5 

16.6 

0.5 

220 

290 

2.9 

2.2 

8.2 

0.3 

33v2 

3.6 

16.5 

0.5 

210 

300 

3.2 

2.5 

8.4 

0.5. 

33-.  0 

3.7 

16.4 

0.6 

200 

310 

3.5 

2.9 

8.7 

0.7 

32.7 

3.9 

16.2 

0.7 

190 

320 

4.0 

3.4 

9.0 

1.0 

32.4 

4.V 

16.1 

0.8 

180 

330 

4.5 

3.9 

9.3 

1.2 

32.0 

4.2' 

15.9 

1.0 

170 

340 

5.1 

4.4 

9.7 

1.6 

31.6 

4A 

1&.7 

1.1 

160 

350 

5.7 

5.1 

10.1 

1.9 

31.1 

4.7 

15.4 

1.3 

150 

360 

6.4 

5.8 

10.6 

2.3 

30.6 

4.9 

15.2 

1.5 

140 

370 

7.1 

6.6 

11.1 

2.7 

30.1 

5.2 

14.9 

1.7 

130 

380 

7.9 

7.4 

11.7 

3.2 

29.4 

5.5 

14.6 

1.9 

120 

390 

8.7 

8.3 

12.2 

3.6 

28.7 

5.8 

14.3 

2.1 

110 

400 

9.6 

9.2 

12.8 

4.1 

28.0 

6.1 

13.9 

2.3 

100 

410 

10.5 

10.1 

13.5 

4.6 

27.3 

6.5 

13.6 

2.5 

90 

420 

11.5 

11.1 

14.1 

5.2 

26.6 

6.8 

13.2 

2.8 

80 

430 

12.5 

12.2 

14.8 

5.7 

25.8 

7.2 

12.9 

3.1 

70 

440 

13.5 

13.2 

15.5 

6.3 

25.0 

7.6 

12.5 

3.3 

60 

450 

14.5 

14.3 

16.2 

6.9 

24.2 

8.0 

12.1 

3.6 

50 

460 

15.6 

15.4 

17.0 

7.5 

23.4 

8.4 

11.7 

3.9 

40 

470 

16.7 

16.5 

17.7 

8.1 

22.6 

8.8 

11.3 

4.1 

30 

480 

17,8 

17.7 

18,5 

8.7 

21.7 

9.2 

10.8 

4.4 

20 

490 

18.-9 

18.8 

19.2 

9.4 

20.9 

9.6 

10.4 

4.7 

10 

500 

20.0 

20.0 

20.0 

10.0 

20.0 

10.0 

10.0 

5.0 

0 

510 

21.1 

21.2 

20.8 

10.6 

19.1 

10.4 

9.6 

5.3 

990 

520 

22.2 

22.3 

21.5 

11.3 

18.3 

10.8 

9.2 

5.6 

980 

530 

23.3 

23.5 

22.3 

11.9 

17.4 

11.2 

8.7 

5.9 

970 

540 

24.4 

24.6 

23.0 

12.5 

16.6 

11.6 

8.3 

6.1 

960 

550 

25.5 

25.7 

23.8 

13.1 

15.8 

12.0 

7.9 

6.4 

950 

560 

26.5 

26.8 

24.5 

13.7 

15.0 

12.4 

7.5 

6.7 

910 

570 

27.5 

27.8 

25.2 

14.3 

14.2 

12.8 

7.1 

6.9 

930 

580 

28.5 

28.9 

25.9 

14.8 

13.4 

13.2 

6.8 

7.2 

920 

590 

29.5 

29.9 

26.5 

15.4 

12.7 

13.5 

6.4 

7.5 

910 

600 

30.4 

30.8 

27.2 

15.9 

12.0 

13.9 

6.1 

7.7 

900 

610 

31.3 

31.7 

27.8 

16.4 

11.3 

14.2 

5.7 

7.9 

890 

620 

32.1 

32.6 

28.3 

16.8 

10.6 

14.5 

5.4 

8.1 

880 

630 

32.9 

33.4 

28.9 

17.3 

9.9 

14.8 

5.1 

8.3 

870 

640 

33.6 

34.2 

29.4 

17.7 

9.4 

15.1 

4.8 

8.5 

860 

650 

34.3 

34.9 

29.9 

18.1 

8.9 

15.3 

4.6 

8.7 

850 

660 

34.9 

35.6 

30.3 

18.4 

8.4 

15.6 

4.3 

8.9 

840 

670 

355 

36.1 

30.7 

18.8 

8.0 

15.8 

4.1 

9.0 

830 

680 

36.0 

36.6 

31.0 

19.0 

7.6 

16.0 

3.9 

9.2 

820 

690 

36.5 

37.1 

31.3 

19.3 

7.3 

16.1 

3.8 

9.3 

810 

700 

36.8 

37.5 

31.6 

19.5 

7.0 

16.3 

3.6 

9.4 

800 

710 

37.1 

37.8 

31.8 

19.7 

6.8 

16.4 

3.5 

9.5 

790 

720 

37.4 

38.1 

32.0 

19.8 

6.5 

16.5 

3.4 

9.5 

780 

730 

37.6 

38.3 

32.1 

19.9 

6.4 

16.5 

3.4 

9.6 

770 

740 

37.7 

,38.4 

32.2 

20.0 

6.3 

16.6 

3.3 

9.6 

760 

750 

37.7 

38.4 

32.2 

20.0 

6.3 

16.6 

3.3 

9.6 

750 

Arg. 

20 

Arg. 

0 

10.0 

500 

10 

10.9 

510 

20 

11.8 

520 

30 

12.7 

530 

40 

13.5 

540 

50 

143 

550  I 

60 

15.0 

560 

70 

15.7 

570 

80 

16.2 

580 

90 

16.7 

590 

100 

17.0 

600 

110 

17.2 

610 

120 

17.4 

620 

130 

17.4 

630 

140 

17.2 

640 

150 

17.0 

650 

160 

16.7 

660 

170 

16,2 

670 

180 

15.7 

680 

190 

15.0 

690 

200 

14.3 

700 

210 

13.5 

710 

220 

12.7 

720 

230 

11.8 

730 

240 

10.9 

740 

250 

10.0 

750 

260 

9.1 

760 

270 

8.2 

770 

280 

7.3 

780 

290 

6.5 

790 

300 

5.7 

800 

310 

5.0 

810 

320 

4.3 

820 

330 

3.8 

830 

340 

3.3 

840 

350 

3.0 

850 

360 

2.8 

860 

370 

2.6 

870 

380 

2.6 

880 

390 

2.8 

890 

400 

3.0 

900 

410 

3.3 

910 

420 

3.8 

920 

430 

4.3 

930 

440 

5.0 

940 

450 

5.7 

950 

460 

6.5 

960 

470 

7.3 

970 

480 

8.2 

980 

490 

9.1 

990 

500 

10.0 

1000 

TABLE  XLVIL 


TABLE  XLVIII.  69 


Equations  21  to  29, 


Equations  30  and  31. 


1 

21 

22 

23 

24 

25 

26 

27 

28 

29 

1 

25 

7.8 

3.2 

7.1 

6.1 

5.9 

4.1 

5.8 

4.3 

5.7 

25 

127 

7.8 

3.2 

7.1 

6.1 

5.9 

4.1 

5.8 

4.3 

5.7 

23 

29 

7.7 

3.3 

7.0 

6.1 

5.9 

1.1 

5.8 

4.3 

5.7 

21 

31 

7.6 

3.3 

7.0 

6.0 

5.8 

4.2 

5.7 

4.3 

5.7 

19 

33 

7.5 

3.4 

6.8 

6,0 

5.8 

4.2 

5.7 

4.4 

5.6 

17 

35 

7.3 

3.5 

6.7 

5,9 

5.7 

4.3 

5.6 

4.4 

5.6 

15 

37 

7.0 

3.7 

6.5 

5.8 

5.7 

4.3 

5.6 

4.5 

5.5 

13 

39 

6.8 

3.9 

6.3 

5.7 

5.6 

4.4 

5.5 

4.6 

5.4 

11 

41 

6.5 

4.0 

6.1 

5.6 

5.5 

4.5 

5.4 

4.6 

5.4 

09 

43 

6.2 

4.2 

5.9 

5.5 

5.4 

4.6 

5.3 

4.7 

5.3 

07 

45 

5.9 

4.4 

5,6 

5.3 

5.3 

4.7 

5.2 

4.8 

5.2 

05 

47 

5.5 

4.7 

5.4 

5.2 

5.2 

4.8 

5.1 

4.9 

5.1 

03 

49 

5.2 

4,9 

5.1 

5.1 

5.1 

4.9 

5.0 

5.0 

5.0 

01 

51 

4.8 

5.1 

4.9 

4.9 

4.9 

5.1 

5.0 

5.0 

5.0 

99 

53 

4.5 

5.3 

4.6 

4.8 

4.8 

5.2 

4.9 

5.1 

4.9 

97 

55 

4.1 

5.6 

4.4 

4.7 

4.7 

5.3 

4.8 

5.2 

4.8 

95 

57 

3.8 

5.8 

4.1 

4.5 

4.6 

5.4 

4.7 

5.3 

4.7 

93 

59 

3.5 

6.0 

3.9 

4.4 

4.5 

5.5 

4.6 

5.4 

4.6 

91 

61 

3.2 

6.1 

3.7 

4.3 

4.4 

5.6 

4.5 

5.4 

4.6 

89 

63 

3.0 

6.3 

3.5 

4.2 

4.3 

5.7 

4.4 

5.5 

4.5 

87 

65 

2.7 

6.5 

3.3 

4.1 

4.3 

5.7 

4.4 

5.6 

4.4 

85 

67 

2.5 

6.6 

3.2 

4.0 

4.2 

5.8 

4.3 

5.6 

4.4 

83 

69 

2.4 

6.7 

3.0 

4.0 

4.2 

5.8 

4.3 

5.7 

4.3 

81 

71 

2.3 

6.7 

3.0 

3.9 

4.1 

5.9 

4.2 

6.7, 

4.3 

79 

73 

2.2 

6.8 

2.9 

3.9 

4.1 

5.9 

4.2 

5.7 

4.3 

77 

75 

2.2 

6.8 

2.9 

3.9 

4.1 

5.9 

4.2 

5.7  J 

4.3 

75 

TABLE  XLIX. 
Equation  432.     Argument,  Supp.  of  Node. 


III* 

IVs 

V« 

VI* 

VII* 

VIII* 

0 

0 

3.1 

4.0 

6.5 

10.0 

13.5 

16.0 

0 

30 

2 

3.1 

4.2 

6.8 

10.2 

13.7 

16.1 

28 

4 

3.1 

4.3 

7.0 

10.5 

13.8 

16.2 

26 

6 

3.1 

4.4 

7.2 

10.7 

14.0 

16.3 

24 

8 

3.2 

4.6 

7.4 

11.0 

14.2 

16.4 

22 

10 

3.2 

4.7 

7.« 

11.2 

14.4 

16.5 

20 

12 

3.3 

4.9 

7.9 

11.4 

14.6 

16.6 

18 

14 

3.3 

5.0 

8.1 

11.7 

14.8 

16.6 

16 

16 

3.4 

5.2 

8.3 

11.9 

15.0 

16.7 

14 

18 

3.4 

5.4 

8.6 

12.1 

15.1 

16.7 

12 

20 

3.5 

5.6 

8.8 

12.4 

15.3 

16.8 

10 

22 

3.6 

5.8 

9.0 

12.6 

15.4 

16.8 

8 

24 

3.7 

6.0 

9.3 

12.8 

15.6 

16.9 

6 

26 

3.8 

6.2 

9.5 

13.0 

15.7 

16.9 

4 

28 

3.9 

6.3 

9.8 

13.2 

15.8 

16.9 

2 

30 

4.0 

6.5 

10.0 

13.5 

16.0 

16.9 

0 

Us 

I» 

O* 

XL? 

X* 

IX* 

Arg 

30 

31 

0 

5.0 

5.0 

2 

5.0 

50 

4 

4.9 

5.1 

6 

4.9 

5.1 

8 

4.8 

5.2 

10 

4.8 

5.2 

12 

4.7 

5.3 

14 

4.6 

5.4 

16 

4.5 

5.5 

18 

4.4 

5.5 

20 

4.2 

5.6 

22 

4.1 

5.7 

24 

4.0 

5.8 

26 

3.9 

5.8 

28 

3.8 

5.9 

30 

3.7 

5.9 

32 

3.7 

5.9 

34 

3.7 

5.9 

36 

3.7 

5.9 

38 

3.8 

5.8 

40 

3.9 

5.7 

42 

4.1 

5.6 

44 

4.3 

5.5 

46 

4.5 

5.3 

48 

4.8 

5.2 

50 

5.0 

5.0 

52 

5.2 

4.8 

54 

5.5 

4.7 

56 

5.7 

4.5 

58 

5.9 

4.4 

60 

6.1 

4.3 

62 

6.2 

4.2 

64 

6.3 

4.1 

66 

6.3 

4.1 

68 

6.3 

4.1 

70 

6.3 

4.1 

72 

6.2 

4.1 

74 

6.2 

4.2 

76 

6.0 

4.2 

78 

5.9 

4.3 

80 

5.8 

4.4 

82 

5.7 

4.5 

84 

5.5 

4.6 

86 

5.4 

4.6 

88 

5.3 

4.7 

90 

5.2 

4.8 

92 

5.1 

4.8 

94 

5.1 

4.9 

96 

5.0 

4.9 

98 

5.0 

5.0 

100 

5.0 

5.0 

Constant  55" 


70 


TABLE   L. 
Evection. 

Argument.     Evection,    corrected. 


0 

I* 

II* 

in* 

IV* 

i 

1 

'i° 

Diff.  2o 

Diff'.!2; 

Diff 

2° 

Diff. 

2° 

Diff 

2° 

Diff 

,     .. 

/    ,/ 

/    ,/ 

/    .. 

M 

,     „ 

,    „ 

0 

1 
9 

3 

4 
5 

30  00.0 
31  25.5 
32  50.  y 
34  16.3 
3541.6 
37    6.7 

85.5 

85.4 

J85.4 
185.3 
85.1 

1043.5 
11  56.7 
13    9.0 
14  20.6 
15  31.3 
1641.1 

73  o  40    9.7 
L*  *  40  50.6 

';•;?  41  30.1 

i*'!;  42    8.3 
'"•'   4245.1 
0  4.320.6 

40.9 
39.5 

138.2 
(36.8 
35.5 

50  25.5 
50  23.5 
5020.1 
50  15.2 
50    8.8 
50    1.0 

2.0 
3.4 
4.9 
6.4 

7.8 

39    8.3 
38  24.9 
37  40.4 
36  54.6 
36    7.6 
35  19.5 

43.4 
44.5 

45.8 
47.0 
48.1 

942.0 
829.3 
716.0 
6    2.0 
447.4 
332.2 

72.7 
73.3 
74.0 
74.6 
75.2 

85.  T 

,69.0  i 

34  1 

93 

49.3 

75.9 

6 

7 
8 
9 
10 

38  SI.S'JMQ 
39  56*7 
4l2L4!ti 
42  45.8  oTo 
44  10.  II84'3 

1750.1 
1858.2 
20    5.3 
21  11.5 
22  16.7 

'rs      43  54.7 
E'-I   4427.4|ff-J 

66:2  4458'829:9 

4951.7 
4941.0 
49  28.8 
49  15.1 
49    0.2 

10.7 
12.2 
13.7 
14.9 

34  30.2 
33  39.7 
3248.1 
31  55.4 
31     1.6 

50.5 
51.6 
52.7 
53.8 

2  16.3 
0  59.9 

76.4 
76.9 

77.4 
78.0 

59  43.0 
58  25.6 
57    7.6 

83.9 

64.3 

272 

16.7 

54.9 

78.4 

11 
12 
18 

14 
15 

45  34.0  oo  7 
4657.7^ 

4944183-0 
<tj  tt.L  g2  6 

2321.0 
24  24.2 
25  26.4 
26  27.6 
27  27.6 

63.2 
62.2 
61.2 
60.0 

46  24.5 
46  B*.8gJ'I 

47  37.4  ??'? 

47  58.8  ^l'4 

48  43.5 
48  25.6 
48    6.3 
4745.5 
17  23.3 

17.9 
19.3 

20.8 

22.2 

30    6.7 

29  10.7 
28  13.7 
27  15.7 
26  16.6 

56.0 
57.0 
58.0 
59.1 

55  49.2 
54  30.3 
53  11.0 
51  51.3 
50  31.2 

78.9  f 
79.3 
79.7 
80.1 

j82.2 

59.0 

200 

235 

60.0 

30.5 

O  CO  CO  -J  O) 

522S.9fl1  « 
53  50.7  8 
55  12.0  SH 
53  32.9  8^  o 

57  53.2 

28  26.6 
29  24.6 
30  21.4 
31  17.0 
32  11.5 

58.0 
56.8 
55.6 
54.5 

48  18.8 
48  37.4 
4854.5 
4910.1 
49  24.4 

18.6 
17.1 
15.6 
14.3 

46  59.8 

46  34.8 
46    8.5 
45  40.7 
45  11.6 

25.0 
26.3 
27.8 
29.1 

25  16.6 
24  15.6 
23  13.6 
22  10.7 
21    6.8 

61.0 
62.0 
62.9 
63.9 

49  10.7 
47  49.9 
46  28.8 
45    7.5 
43  45.8 

80.8 
81.1 
81.3 
81.7 

79.8 

53.3 

127 

30.4 

64.7 

81.9 

21 

24 
,25 

59  13.0  79  ^ 
"0  32.3  78;'7 
151.0  78  1 
3    9.1774 
426.5, 

33    4.8 
33  57.0 
34  47.9 
35  37.7 

36  26.2 

52.2 
50.9 
49.8 

48.5 

4937.1 
4948.3 
4958.1 
50    6.4 
50  13.3 

11.2 
9.8 
8.3 
6.9 

4441.2 
44    9.5 
43  36.4 
43    1.9 

42  26.2 

31.7 
33.1 
34.5 
35.7 

20    2.1 
18  56.4 
1749.9 
16  42.6 
15  34.4 

K.  „  42  23.9 

*i  41  i-s 

£X'£  39  39.5 
SI'*  38  17.0 
OB"*  3654.4 

82.1 
82.3 
82.5 
82.6 

76.8 

47.2 

5.4 

37.0 

68.9  1 

82.7 

2G 
27 

30 

5  43.3 
6  59.4 
8  14.9 
9  29.6 
1043.5 

76.1  1 
75.5 
74.7 
73.9 

37  13.4 
37  59.4 
38  44.2 
39  27.6 
40    9.7 

46.0 
44.8 
43.4 
42.1 

50  18.7 
50  22.6 
50  25.0 
50  26.0 
5025.5 

3.9 

2.4 

1.0 

41  49.2 
41  10.8 
4031.2 
39  50.4 
39    8.3 

38.4 
39.6 
40.8 
42.1 

425.5 
3  15.7 
2    5.2 
10  54.0 
942.0 

69.8 
70.5 
71.2 
72.0  , 

3531.7 
34    8.8 
32  45.9 
31  23.0 
30    0.0 

82.9 
82.9 
82.9 
83.0 

2° 

2° 

2° 

ia° 

2° 

1° 

TABLE   L. 
Evection. 

Argument.     Evection,  corrected. 


71 


VI* 

|  VII* 

VIII* 

IX* 

X* 

XI* 

s? 

1° 

Diff.O0 

Diff.'o0 

Diff.O0 

Diff.  0^ 

Diff. 

<*> 

Diff. 

•q 

0 
1 

0 

3 

4 
5 

30    00 

23  37.0 
27  14.1 
25  51.2 
24  28.3 
23    5.6 

83.0 
82.9 
82.9 
829 
82.7 

50  18.0 
49  6.0 

47  54.8 
46  44.3 
45  34.5 
44  25.6 

72.0 
71.2 
70.5 
69.8 
68.9 

2051.7 
20    9.6 
19  28.8 
1849.2 
18  10.8 
1733.8 

42.1 
40.8 
39.6 
38.4 
37.0 

9  34.5 
9  34.0    J;J 
935.0    i? 
937.4    l\ 
941.3    J.'J 
946.7    5'4 

19  50.3 
20  32.4 

21  15.8 
22  0.6 
22  46.6 
23  33.8 

42.1 
43.4 
44.8 
46.0 
47.2 

49  16.5 
50  30.4 
5145.1 
53    0.6 
54  16.7 
55  33.5 

73.9 

74.7 
75.5 
76.1 
76.8 

82.6 

68.2 

35.7 

6.9 

48.5 

77.4 

6 

7 
8 
9 
10 

21  43.0 
20  20.5 
IS  58.2 
1738.1 
16  14.2 

82.5 
82.3 
82.1 
81.9 

43  17.4 
42  10.1 
41  3.6 
39  57.9 
38  53.2 

67.3 
66.5 
65.7 
64.7 

16  58.1 
16  23.6 
15  50.5 
15  18.8 
14  48.4 

34.5 
33.1 
31.7 
30.4 

953.6 
10    1.9    «-3 
1011.7    ^ 
!«»*{*•; 
10  35.6  u 

24  22.3 

25  12.1 
26  3.0 
26  55.2 
27  48.5 

49.8 
50.9 
52.2 
53.3 

56  50.9  7fi  , 
58    9.0  ™'i 
59  27.7  ?H 
04770^ 
2    6.8 

81  7 

639 

29.1 

143 

545 

80.3 

11 
12 

13 
14 
15 

14  52.5 
1331  2 
12  10.1 
10  49.3 
928.8 

81.3 
81.1 

80.8 
80.5 

3749.3 
36  46.4 
35  44.4 
34  43.4 
33  43.4 

62.9 
62.0 
61.0 
60.0 

14  19.3 
1351.5 
1325.2 
13    0.2 
1236.7 

27.8 
26.3 
25.0 
23.5 

1049.9 
11    5.5 
11  22.6 
1141.2 
12    1.2 

15.6 
17.1 
18.6 
20.0 

28  43.0 
29  38.6 
30  35.4 
31  33.4 
32  32.4 

55.6 
56.8 
58.0 
59.0 

327.1 
448.0 
6    9.3 
731.1 
853.3 

80.9 
81.3 
81.8 
82.2 

80.1 

59.1 

22.2 

21.4 

600 

82.6 

16 

17 
IS 
19 
20 

8    8.7 
649.0 
529.7 
4  10.8 
252.4 

79.7 
79.3 
78.9 

78.4 

32  44.3 
31  46.3 
30  49.3 
29  53.3 
28  58.4 

58.0 
57.0 
56.0 
54.9 

12  14.5 
11  53.7 
11  34,4 
11  16.5 
10  59.8 

20.8 
19.3 
17.9 
16.7 

1222.6 
12  45.5 
13    9.8 
13  35.5 
14    2.7 

22.9 
24.3 
25.7 

27.2 

33  32.4 
34  33.6 
35  35.8 
36  39.0 
37  43.3 

61.2 
62.2 
63.2 
64.3 

10  15.9 
1138.9 
13    2.3 
1426.0 
1549.9 

83.0 
83.4 
83.7 
83.9 

78.0 

53.8 

14,9 

28.6 

65.2 

84.3 

21 
22 
23 
24 
25 

1  34.4 
0  17.0 

77.4 
76.9 
76.4 
75.9 

28  4.6 
27  11.9 
26  20.3 
25  29.8 
24  40.5 

52.7 
51.6 
50.5 
49.3 

10  44.9 
1031.2 
10  19.0 
10    8.3 
9  59.0 

13.7 
12.2 
10.7 
9.3 

1431.3 

15    1.2 
15  32.6 
16    5.3 
1639.4 

29.9 
31.4 
32.7 
34.1 

38  48.5 
39  54.7 
41  1.8 
42  9.9 
43  18.9 

66.2 
67.1 
68.1 
69.0 

1714.2 
1838.eC'* 

20    3.3^ 
21  28.2  f'9 
2253.3i 

59    0.1 
57  43.7 

56  27.8 

. 

75.2 

48.1 

7.8 

355 

698 

85.1 

26 

27 
28 
29 
30 

55  12.6 
53  58.0 
52  44.0 
51  30.7 
50  18.0 

74.6 
74.0 
73.3 
72.7 

23  52.4 
23  5.4 
22  19.6 
21  35.1 
2051.7 

47.0 
45.8 
44.5 
43.4 

951.2 
944.8 
939.9 
936.5 
934.5 

6.4 
4.9 
3.4 
2.0 

17  14.9 
1751.7 
18  29.9 
19    9.4 
19  50.3 

36.8 
38.2 
39.5 
40.9 

4428.7 
45  39.4 
4651.0 
48  3.3 
49  16.5 

70.7 

71.6 
723 
73.2 

24  18.4 
25  43.7 
27    9.1 
28  34.5 
30    0.0 

85.3 

85.4 
85.4 
85,5 

0° 

0° 

0° 

0° 

0° 

1° 

72 


TABLE   LI, 

Equation  of  Moon's  Centre. 
Argument.     Anomaly   corrected. 


0* 

I* 

II* 

III* 

IV* 

V* 

7° 

Diff 
for  10 

10° 

Diff 
for  10 

12° 

Diff 
forlO 

13° 

Diff 
for  10 

12° 

Diff 
for  10 

9° 

Diff 
for  10 

0    ' 

0  0 
30 
1  0 
30 
2  0 
30 

0   0.0 

332.6 
7   5.2 
1037.8 
1410.3 
1742.7 

70.9 

709 
70.9 

70.8 
70.8 

2057.9 
2355.6 
26  52.2 
2947.7 
32  42.0 
3535.2 

59.2 

58:9 

58.5 
58J 
57.7 

3843.6 
40  14.0 
4142.7 
43    9.6 
4434.9 
4558.4 

30.1 
29.6 
29.0 

28.4 
27.8 

1735.2 
17209 
17   4.8 
1647.1 
1627.6 
16    6.5 

K 

5.9 
6.5 
7.0 

1620.8 
1435.3 
1248.5 
11    0.4 
911.1 
720.5 

// 
35.2 
35.6 
36.0 
36.4 
36.9 

5828.9 
5543.8 
5258.0 
5011.6 
4724.5 
4436.8 

// 
55.0 
55.3 
55.5 
55.7 
55.9 

70.8 

57.3 

27.3 

7.6 

37.3 

56.1 

3  0 
30 
4  0 
30 
5  0 

21  15.0 
24  47.3 
2819.4 
3151.2 
35  23.0 

70.8 
70.7 
70.6 
70.6 

3827.1 
41  18.0 
44   7.6 
46  56.0 
49  43.2 

57.0 
56.5 
56.1 
55.7 

47  20.2 
48  40.3 
4958.7 
51  15.3 
5230.2 

26.7 
26.1 
25.5 
25.0 

1543.7 
1519.2 
1453.1 
1425.2 
1355.8 

8.2 
8.7 
9.3 
9.8 

528.7 
335.6 
141.3 

37.7 
38.1 
38.5 
38.9 

41  48.5 
3859.5 
36  10.0 
33  19.8 
3029.1 

56.3 
56.5 
56.7 
56.9 

5945.8 
5749.1 

70.5 

55.3 

24.4 

10.4 

39.3 

57.1 

30 
6  0 
30 
7  0 
30 

3854.5 
42  25.8 
4556.9 
4927.7 
52  58.2 

70.4 
70.4 
70.3 
70.2 

5229.1 
55  13.8 
5757.2 
~039L3 
320.1 

54.9 
54.5 
54.0 
53.6 

5343.3 
5454.7 
56   4.4 
57  12.3 

58  18.5 

23.8 
23.2 
22.6 
22.1 

1324.7 
1251.9 
1217.4 
1141.4 
11    3.7 

10.9 
11.5 
12.0 
12.6 

5551.1 
53  52.0 
5151.7 
49  50.3 
4747.6 

39.7 
40.1 
40.5 
40.9 

27  37.8 
24  45.9 
21  53.5 
19    0.6 
16    7.1 

57.3 
57.5 
57.G 
57.8 

70.1 

53.2 

21.5 

13.1 

41.3 

58.0 

8  0 
30 
9  0 
30 
10  0 

56  28.5 
59  58.4 
~3~2lM) 
657.2 
1026.0 

70.0 
69.9 
69.7 
69.6 

559.7 
837.9 
11  14.8 
1350.3 
1624.5 

52.7 
52.3 

51.8 
51.4 

59  22.9 
025.6 
126.5 
225.7 
323.0 

20.9 
20.3 
19.7 
19.1 

1024.3 
943.4 
9    0.8 
816.6 
730.8 

13.6 
14.2 
14.7 
15.3 

4543.8 
43  38.9 
41  32.8 
39  25.6 
37  17.3 

41.7 
42.0 
42.4 
42.  "8 

1313.1 

1018.6 
723.6 
428.1 
132.2 

58.2 
58.3 
58.5 
58.6 

69.5 

50.9 

18.6 

15.8 

43.1 

58.8 

30 
11  0 
30 
12  0 
30 

1354.5 

1722.5 
2050.1 
2417.3 
2744.0 

69.3 
69.2 
69.1 
68.9 

18  57.3 
21  28.8 
23  58.8 
2627.5 
28  54.7 

50.5 
50.0 
49.6 
49.1 

418.7 
5  12.5 
6   4.6 
654.9 
743.5 

17.9 
17.4 
16.8 
16.2 

6  43.4 
554.4 
5   3.9 
411.7 
318.0 

16.3 
16.8 
17.4 
17.9 

35    7.9 

3257.4 
3045.8 
2833.1 
26  19.4 

43.5 
43.9 
44.2 
44.6 

58  35.8 
5538.9 
5241.7 
4943.9 
4645.8 

59.0 
59.1 
59.3 
59.4 

68.7 

48.6 

15.6 

18.4 

44.9 

!59.5 

13  0 
30 
14  0 
30 
15  0 

31  10.2 
34  35.8 
38    1.0 
41  25.6 
4449.6 

68.5 
68.4 
68.2 
68.0 

3120.5 
3344.9 
36    7.9 
38  29.4 
4049.3 

48.1 
47.7 
47.2 
46.6 

830.3 
915.4 
958.6 
1040.1 
11  19.9 

15.0 
14.4 
13.8 
13.3 

222.7 
125.8 
027.4 

19.0 
19.5 
20.0 
20.5 

24   4.6 

2148.8 
1931.9 
1714.1 
1455.2 

45.3 
45.6 
45.9 
46.3 

4347.3 
4048.4 
3749.1 
3449.5 
31494 

59.6 
59.8 
59.9 
60.0  | 

59  27.4 
5825.9 

8° 

11° 

13° 

12° 

11° 

|8° 

TABLE  LI. 

Equation  of  Moon's  Centre. 
Argument.      Anomaly  corrected. 


73 


VI* 

' 

VII* 

VIII* 

IX* 

x« 

XI» 

Diff 

DifF 

Diff 

Diff   ,0 

Diff 

oO 

Diff 

7° 

for  10 

4° 

forlO 

1° 

for  10 

for  10  1 

for  10 

O 

for  10 

O       ' 

0  0 
30 
1  0 
30 
2  0 
30 

0   0.0 
5654.6 
5349.2 
5043.9 
4738.6 
4433.4 

61.8 
61.8 
61.8 
61.8 
61.7 

/     rr 

131.1 
5846.7 
56    3.0 
5320.0 
50  37.7 
4756.2 

54.8 
54.6 
543 
54.1 
53.8 

43  39.2 
41  55.0 
40  12.0 
3830.5 
36  50.3 
3511.3 

34.7 
34.3 
33.8 
33.4 
33.0 

/    // 
42  24.8 
4212.1 
42    1.2 
41  52.0 
41  44.4 
41  38.7 

4.2 
3.6 
ai 
2.5 
1.9 

2116.4 

2248.5 
2422.2 
2557.7 
2734.8 
29  13.7 

30.7 
31.2 
31.8 
32.4 
33.0 

39    2.1 
42    0.8 
45   0.7 
48    1.7 
51    3.7 
54   6.7 

59.6 
60.0 
60.3 
60.7 
61.0 

61.8 

53.6 

32.5 

1.4 

33.5 

61.3 

3  0 
30 
4  0 
30 
5  0 

4128.1 
38  23.0 
35  18.0 
32  13.0 
29    8.1 

61.7 
61.7 
61.7 
61.6 

45  15.4 
4235.3 
3956.0 
3717.4 
3439.6 

53.4 
53.1 
52.9 
52.0 

3333.7 
31  57.5 
3022.6 
2849.0 
2716.8 

32.1 
31.6 
31.2 
30.7 

41  34.6 
41  32.2 
4131.6 
41  32.7 
41  35.6 

0.8 
0.2 
0.4 
1.0 

30  54.2 
3236.3 
3420.2 
36    5.6 
3752.8 

34.0 
34.6 
35.1 
35.7 

57  10.7 
~OT5^ 
321.8 
628.8 
936.8 

61.7 
62.0 
62.3 
62.7 

61.6 

52.3 

30.2 

1.5 

36.2 

63.0 

30 
6  0 
30 
7  0 
30 

26   3.4 

2258.8 
1954.3 
1650.0 
1345.8 

61.5 
61.5 
61.4 
61.4 

32    2.7 
2926.5 
2651.1 
2416.6 
2142.9 

52.1 
51.8 
51.5 
51.2 

2546.1 
2416.7 

2248.7 
2122.1 
1956.9 

29.8 
29.3 
28.9 

28.4 

4140.1 
4146.4 
41  54.5 
42   4.3 
4215.9 

2.1 
2.7 
3.3 
3.9 

3941.5 
4132.0 
43  24.0 
45  17.7 
4712.9 

36.8 
37.3 
37.9 

38.4 

1245.7 
1555.5 
19    6.2 
22  17.8 
25  30.3 

63.3 
63.6 
63.9 
64.2 

61.3 

51.0 

27.9 

4.4 

39.0 

64.5 

8  0 
30 
9  0 
30 
10  0 

1041.9 
738.0 
434.4 
131.0 

58271 

61.3 
61.2 
61.1 
61.1 

1910.0 
163S.O 
14   6.9 
1136.6 
9    7.3 

50.7 
50.4 
50.1 
49.8 

1833.1 
1710.8 
1549.8 
1430.4 
1312.5 

27.4 
27.0 
265 
26.0 

4229.2 
42  44.2 
43    1.1 
4319.6 
4339.9 

5.0 
5.6 
6.2 

6.8 

49    9.8 
51    8.3 
53   8.4 
5510.1 
5713.3 

39.5 
40.0 
40.6 
41.1 

2843.7 
31  57.8 
3512.9 
3828.7 
41  45.2 

64.7 
65.0 
65.3 
65.5 

61.0 

49.5 

25.5 

7.4 

41.6 

65.8 

30 
11   0 
30 
12  0 
30 

5524.9 
5222.2 
4919.7 
46  17.5 
43  15.6 

60.9 
60.8 
60.7 
60.6 

638.9 
411.3 
144.7 

49.2 
48.9 
48.6 

48.2 

1155.9 
1040.9 
927.3 

815.2 
7   4.6 

25.0 
24.5 
24.0 
23.5 

44   2.0 

4425.9 
4451.5 
4518.8 
4548.0 

8.0 
8.5 
9.1 
9.7 

59  18.2 
124T5 
332.4 
541.9 
752.9 

42.1 
42.6 
43.2 
43.7 

45   2.6 

4820.7 
5139.6 
5459.1 
5819.3 

66.0 
66.3 
66.5 
66.7 

59  18.9 
56  54.2 

60.5 

47.9 

23.1 

103 

442 



67.0 

13  0 
30 
14  0 
30 
15  0 

4014.0 
3712.6 
3411.6 
31  10.9 
28  10.6 

60.5 
60.3 
60.2 
60.1 

5430.4 
52    7.5 
4945.6 
4724.7 
45   4.8 

47.6 
47.3 
47.0 
46.6 

555.4 
447.8 
341.7 
237.1 
134.1 

22.5 
22.0 
21.5 
21.0 

46  18.9 
4651.5 
4726.0 
48    2.2 
4840.1 

10.9 
11.5 
12.1 
12.6 

10   5.5 
1219.5 
1435.1 
1652.1 
1910.7 

44.7 
45.2 
45.7 
46.2 

1  40.3 
5    1.9 

824.1 
1146.9 
15104 

67.2 
67.4 
67.6 
67.8 

53 

2° 

1° 

0° 

i 

5° 

74 


TABLE   LI. 

Equation  of  Moon's  Centre. 
Argument.     Anomaly  corrected. 


0* 

I* 

II* 

III* 

IV* 

V* 

8° 

Diff 
forlO 

11° 

Diff 
forlO 

13° 

Diff 
forlO 

12° 

Diffl   10 
forlOn 

Diff 
forlO 

8° 

Diff 
for  10 

O    ' 

/    // 

/    // 

/    ,' 

/     // 

/     ., 

,    „ 

15  0 
30 
16  0 
30 
17  0 
30 

4449.6 
4813.1 
5135.9 
5458.1 
5819.7 

67.8 
67.6 
67.4 
67.2 
67.0 

4049.3 
43    7.9 
4524.9 
4740.5 
4954.5 
52    7.1 

46.2 
45.7 
45.2 

44.7 
44.2 

1119.9 

1157.8 
1234.0 
13  8.5 
1341.1 
1412.0 

12.6 
12.1 
11.5 
10.9 
10.3 

5825.9 
5722.9 
56  18.3 
5512.2 
54  4.6 
5255.4 

21.0 
21.5 
22.0 
22.5 
23.1 

1455.2 
1235.3 
1014.4 
752.5 
529.6 
3   5.8 

46.6 
47.0 
47.3 
47.7 
47.9 

3149.4 
2849.1 
2548.4 
22  47.4 
1946.0 
1644.4 

60.1 
f.0.2 
60.3 
60.5 
60.5 

140  7 

66.7 

437 

9.7 

23.5 

483 

60.6 

18  0 
30 
19  0 
30 
20  0 

5    0.9 
820.4 
1139.3 
1457.4 
1814.8 

66.5 
66.3 
66.0 
65.8 

5418.1 
5627.6 
5835.5 
~6~4T8 
246  7 

43.2 

42.6 
42.1 
41.6 

1441.2 
15  8.5 
1534.1 
1558.0 
1620.1 

9.1 

8.5 
8.0 
7.4 

5144.8 
5032.7 
4919.1 
48  4.1 
46  47.5 

24.0 
24.5 
25.0 
25.5 

041.1 
58  15.3 
5548.7 
5321.1 
5052.7 

48.6 
48.9 
49.2 
49.5 

1342.5 
1040.3 
737.8 
435.1 
132.2 

60.7 
60.8 
60.9 
61.0 

65.5 

41.1 

6.8 

26.0 

49.8 

61.1 

30 
21  0 
30 
22  0 
30 

2131.3 

24  47.  1 
28    2.2 
31  16.3 
3429.7 

65.3 
65.0 
64.7 
64.5 

449.9 
651.6 
851.7 
1050.2 
1247.1 

40.6 
40.0 
39.5 
39.0 

1640.4 
1658.9 
1715.8 
1730.8 
1744.1 

6.2 
5.6 
5.0 
44 

45  29.6 
4410.2 
4249.2 
4126.9 
40  3.1 

26.5 
27.0 
27.4 
27.9 

48  23.4 
4553.1 
4322.0 
4050.0 

3817.1 

50.1 
50.4 
50.7 
51.0 

58  29.0 
5525.6 
5222.0 
4918.1 
46  14.2 

61.1 
61.2 
61.3 
61.3 

64.2 

38.4 

3.9 

28.4 

51.2 

61.4 

23  0 
30 
24  0 
30 
25  0 

3742.2 
4053.8 
44   4.5 
4714.3 
50  23.2 

63.9 
63.6 
63.2 
63.0 

1442.3 
1636.0 
1828.0 
2018.5 

22    7.2 

37.9 
37.3 

36.8 
36.2 

1755.7 
18  5.5 
1813.6 
1819.9 
18  24.4 

33 
2.7 
2.1 
1.5 

33  37.9 
3711.3 
35433 
34  13.9 
3243.2 

28.9 
29.3 
29.8 
30.2 

3543.4 
33   8.9 
3033.5 
2757.3 
2520.4 

51.5 
51.8 
52.1 
52.3 

43  10.0 
40    5.7 
37    1.2 
3356.6 
3051.9 

61.4 
61.5 
61.5 
61.6 

62.7 

35.7 

*•'  , 

1.0 

307 

526 

61  6 

30 
26  0 
30 
27  0 
30 

5331.2 
5633.2 
59  44.2 

62.3 
R2.0 
61.7 
61.3 

23  54.4 
25  39.8 
2723.7 
29    5.8 
3046.3 

35.1 
34.6 
34.1 
33.5 

1827.3 
IS  28.4 
1827.8 
1825.4 
1821.3 

0.4 
0.2 
0.8 
1.4 

31  11.0 
29  37.4 
28  2.5 
26  26.3 

2448.7 

31.2 
31.6 
32.1 
32.5 

2242.6 
20    4.0 
1724.7 
1444.7 
12    3.8 

52.9 
53.1 
3.3 
53.6 

2747.0 
2442.0 
21  37.0 
1831.8 
1526.6 

61.7 
61.7 
61.7 
61.7 

249.3 
5  53.3 

61.0 

33.0 

1.9 

33.0 

53.8 

61,7 

28  0 
30 
29  0 
30 
30  0 

856.3 
1158.3 
1459.3 
1759.2 
2057.9 

60.7 
60.3 
60.0 
59.6 

32  25.2 
34   2.3 
35  37.8 
3711.5 
3843.6 

32.4 
31.8 
31.2 
30.7 

1815.6 
18  8.0 
1758.8 
1747.9 
1735.2 

2.5 
3.1 
3.6 
4.2 

23  9.7 
21  29.5 
1948.0 
18  5.0 
1620.8 

33.4 
33.8 
34.3 
34.7 

922.3 
640.0 
357.0 
1  13.3 

54.1 
54.3 
54.6 

54.8 

1221.4 
916.1 
610.8 
3   5.4 
0   0.0 

61.8 
61.8 
61.8 
61.8 

5828.9 

!10° 

12° 

13° 

12° 

9° 

7° 

TABLE  LI. 

Equation  of  Moon's  Centre. 
Argument.      Anomaly  corrected. 


75 


VI* 

VII* 

VIII* 

IX* 

X* 

XI* 

Diff 

Diff    o 

Diff 

Diff  Lo 

Diff 

Diff 

5° 

forlO 

2° 

for  10,           * 

1 

for  10 

0° 

for  10 

x 

for  10 

5° 

forlO 

0      / 

,    ,, 

,    „ 

/    // 

,   „ 

,   ,, 

/    „ 

15  0 
30 
16  0 
30 
17  0 
30 

28  10.6 
2510.5 
2210.9 
1911.6 
1612.7 
1314.2 

60.0 
59.9 
59.8 
59.6 
59.5 

45   4.8 
4245.9 
4028.1 
3811.2 
3555.4 
3340.6 

46.3 
45.9 
45.6 
45.3 
44.9 

134.1 
032.6 
59~32l3 
5834.2 
5737.3 
56  42  0 

20.5 
20.0 
19.5 
19.0 
18.4 

4840.1 
49  19.9 
50    1.4 
5044.6 
5129.7 
5216.5 

13.3 

13.8 
14.-4 
15.0 
15.6 

1910.7 
21  30.6 
2352.1 
2615.1 
28  39.5 
31    5.3 

46.6 

47.2 
47.7 
48.1 
48.6 

15  10.4 
8  34.4 
21  59.0 
2524.2 
28  49.8 
3216.0 

68.0 
68.2 
68.4 
68.5 
68.7 

594 

44.6 

J17.9 

16.2 

49.1 

68.9 

18  0 
30 
19  0 
30 
20  0 

1016.1 
718.3 
421.1 
124.2 

59.3 
59.1 
59.0 

58.8 

31  26.9 
29  14.2 
27    2.6 
2452.1 

2242.7 

44.2 
43.9 
43.5 
43.1 

5548.3 
5456.1 
54   5.6 
5316.6 
5229.2 

17.4 
16.8 
16.3 
15.8 

53    5.1 
53  55.4 
5447.5 
5541.3 
56  37.0 

16.8 
17.4 
17.9 
18.6 

3332.5 
36    1.2 

3831.2 
41    2.7 
43  35.5 

!49.6 
50.0 
50.5 
50.9 

3542.7 
39    9.9 
4237.5 
46    5.5 
49  34.0 

69.1 
69.2 
69.3 
69.5 

58278 

58.6 

42.8 

15.3 

19.1 

51.4 

69.6 

305531.9 
21  0,5236.4 
304941.4 
22  04646.9 
304352.9 

58.5 
58.3 

58.2 
58.0 

2034.4 
1827.2 
1621.1 
1416.2 
1212.4 

42.4 
42.0 
41.6 
41.3 

51  43.4 
5059.2 
50  16.6 
4935.7 
48  56.3 

14.7 
14.2 
13.6 
13.1 

5734.3 
58  33.5 
59  34.4 
03771 
141  5 

19.7 
20.3 
20.9 
21.5. 

46    9.7 
48  45.2 
5122.1 
54    0.3 
5639.9 

51.8 
52.3 
52.7 
53.2 

53    2.8 
5631.9 

1TT76 

331.5 
7    1.8 

69.7 
69.9 
70.0 
70.1 

57.8 

40.9 

12.6 

22.1 

53.6 

70.2 

23  0 
30 

24  0 
30 
25  0 

4059.4 
38    6.5 
3514.1 

32  22.2 
2930.9 

57.6 
57.5 
57.3 
57.1 

10    9.7 
8    8.3 
6    8.0 
4   8.9 
210.9 

40.5 
40.1 
39.7 
39.3 

4818.6 
4742.6 
47   8.1 
4635.3 
46   4.2 

12.0 
11.5 
10.9 
10.4 

247.7 
355.6 
5   5.3 

616.7 

729.8 

22.6 
23.2 
23.8 

24.4 

5920.7 

54.0 
54.5 
54.9 
55.3 

1032.3 
14    3.1 
1734.2 
21    5.5 
2437.0 

70.3 
70.4 
70.4 
70.5 

2    2.8 
446.2 
730.9 
1016.8 

j  56.9 

I 

38.9 

9.8 

25.0 

55.7 

70.6 

30 
26  0 
30 
27  0 
30 

2640.2' 
2350.0  .;?!!•' 

21  &5  Sr2 

181L5i561 
15  23.2  5fU 

014.2 

38.5 
38.1 
37.7 
37.3 

45  34.8 
45    6.9 
4440.8 
4416.3 
43  53.5 

9.3 

8.7 
8.2 
7.6 

844.7 
10    1.3 
11  19.7 
1239.8 
14    1.6 

25.5 
26.1 

26.7 
27.3 

13    4.0 
1552.4 
1842.0 
2132.9 

2424.8 

56.1 
56.5 
57.0 
57.3 

28    8.8 
3140.7 
35  12.8 
3845.1 
4217.3 

70.6 
70.7 
70.7 
70.8 

58  18.7 
5024.4 
J431.3 
5239.5 

55.9 

36.9 

7.0 

27.8 

57.7 

70.8 

28  0 
30 
29  0 
SO 
dO  0 

1235.5 
948.4 
7   2.0 
416.2 
131.1 

55.7 
55.5  | 
55.3 
55.0 

5048.9 
48  59.6 
4711.5 
45  24.7 
4339.2 

36.4 
36.0 
35.6 
35.2 

4332.4 
4312.9 
4255.2 
4239.1 

4224.8 

6.5 
5.9 
5.4 
4.8 

1525.1 
1650.4 
1817.3 
1946.0 
21  16.4 

4  27  18.0 

AiO.rr    o/\  -if)  o: 

29.0  !             1 

29'6  36   44 
3(U  39   il 

58.1 
58.5 
58.9 
59.2 

4549.7 
49  22.2 
52  54.8 
56  27.4 
0   0.0 

70.8 
70.9 
70.9 
70.9 

4° 

1° 

0° 

1° 

3° 

7° 

76 


TABLE   LII. 

Variation. 
Argument.     Variation,  corrected. 


0* 

I* 

II* 

111^ 

iv« 

V« 

fo° 

DiffJl0 

Diff. 

1° 

Diff.  0° 

Diff. 

0° 

Diff. 

0° 

Diff. 

0 

1 

2 
3 
4 
5 

I 

L 

38    0.0 
39  13.3 
40  26.5 
|41  39.5 
J42  52.2 
44    4.5 

73.3 
73.3 
73.0 
72.7 
72.3 

8    1.5 
835.5 
9    7.2 
936.5 
10    3.4 
10  27.9 

34.0 
31.7 
29.3 
26.9 
24.5 

6  57.9    g 
6  18.0^-J 
5  35.9  f'1 

45L7Ir1 

A     K  K.46.2 
4.S  2 
3  17.3<48^ 

35  54.4 
34  40.4 
33  26.6 
32  13.0 
30  59.6 
29  46.7 

74.0 

73.8 
73.6 
73.4 
72.9 

529.5 
454.2 
421.3 
350.6 
322.3 
256.5 

35.3 

32.9 
30.7 
28.3 
25.8 

6    1.6 
641.6 
723.9 
8    8.4 
8  55.0 
943.7 

// 
40.0 
42.3 
44.5 
46.6 
48.7 

71.9 

22.0 

50.1 

72.4! 

23.4 

50.8 

6 
7 
8 
9 
10 

45  16.4 
46  27.7 
47  38.4 
48  48.3 
49  57.4 

71.3 
70.7 
69.9 
69.1 

10  49.9 
11    9.4 
11  26.4 
11  40.9 
11  52.9 

19.5 
17.0 
14.5 
12.0 

2  27.2  -    ( 

1  35.3  ^i; 
JLiLSSs 

5946.15?1 
58490 

28  34.3 
27  22.4 
26  11.2 
25    0.7 
2351.1 

233.1 
712  215U 

1  L.&     -t    eo  r/ 

7ft  *  1  1  53.7 

/U.O  |  ,  07  Q 
fiq  R  j  1  d7.8 

by'b   124.5 

21.0 
18.4 
15.9 
13.3 

10  34.5 
1  1  27.3 
12  22.0 
13  18.6 
14  16.9 

52.8 
54.7 
56.6 
58.3 

68.2 

9.3 

58.8 

68.8 

10.8 

60.1 

11 
12 
13 
14 
15 

51    5.6 
52  12.8 
53  18.9 
54  23.8 
55  27.5 

67.2 
66.1 
64.9 
63.7 

12    2.2 
12    9.0 
12  13.2 
12  14.8 
12  13.9 

6.8 
4.2 
1.6 
0.9 

57  50.2 
56  50.0 
55  48.3 
54  45.2 
53  40.9 

60.2 
61.7 
63.1 
64.3 

22  42.3 
21  34.5 
20  27.9 
19  22.3 
18  18.0 

67.8   \  ™ 

s-;  '  ™ 

6430570 
64  3  0  56.7 

8.2 
5.5 
3.0 
0.3 

15  17.0 
16  18.7 
1722.0 
1826.9 
19  33.1 

61.7 
63.3 
64.9 
66.2 

62.3 

3.6 

65.6 

63.0 

2.3 

67.6 

16 

17 
18 
19 
20 

56  29.8 
57  30.7 
5830.1 
59  28.0 

60.9 
59.4 
57.9 
56.2 

12  10.3 
12    4.2 
11  55.5 
11  44.2 
11  30.5 

6.1 
8.7 
11.3 
13.7 

52  35.3 

51  28.5 
50  20.7 
49  11.9 
48    2.2 

66.8 
67.8 
68.8 
69.7 

17  15.0 
16  13.4 
15  13.2 
14  14.6 
13  17.5 

61.6 
60.2 
58.6 
57.1 

059.0 
1    3.9 
1  11.5 
1  21.6 
134.4 

4.9 
7.6 
10.1 

12.8 

20  40.7 
21  49.6 
22  59.6 
24  10.8 
25  22.9 

68.9/ 
70.0 
71.2 
72.1 

0  24  2 

54.5 

16.4 

70.5 

55.3 

15.4 

73.0 

21 
22 
23 
24 
25 

1  18.7 
2  11.4 
3    2.3 
351.2 
438.2 

52.7 
50.9 
48.9 
47.0 

11  14.1 
10  55.3 
10  34.0 
10  10.2 
944.0 

18.8 
21.3 
23.8 
26.2 

46  51.  7  _  „ 
4540.51^ 

4428-6!725 
481«.l{™9 

42    3.2i72'9 

12  22.2 
11  28.5 
1036.7 
9  46.8! 

8  58.81 

53.7 
51.8 
49.9 
48.0 

1  49.8 
2*  7.8 
2  28.3 
251.4 
3  16.9 

18.0 
20.5 
23.1 
2o.5 

26  35.9 
27  49.8 
29    4.5 
30  19.7 
31  35.6 

73.9 

74.7 
75.2 

75.91 

44.9 

28.G  !              |73.3 

146.1 

28.1 

76.3 

26 
27 
28 
29 
30 

523.1 
6    6.0 
646.7 
725.2 
8    1.5 

42.9 
40.7 
38.5 
36.3 

9  15.4 
844.5 
8  11.2 
735.7 
6  57.9 

30.9 
33.3 
35.5 

37.8 

4049.9!7q7 
39  36.2  £  ' 
38  ftt.4  X  A 

o-r     o  A  74.0 

37    8.474Q 
35  54.4  ™ 

8  12.7! 
728.7 
646.8 
6    7.1 
529.5 

A  i  n  3  45-° 
44.0  \A  1K  c 

41  q  !4  15  6 

39  7  i4  48'5 
o?  I  5  23.9 
«j7.b  /.     ,  c 
16    1.6 

30.6 
32.9 
35.4 
37.7 

3251.9 
34   8.6 
35  25.6 
36  42.7 
38    0.0 

76.7 
77.0 
77.1 
77.3 

— 

1° 

1° 

0° 

loo 

1? 

0° 

TABLE  LII. 
Variation. 

Argument.     Variation  corrected. 


1    VI* 

VII* 

. 

VIII* 

IX* 

X* 

XI* 

'  J 

1 

0° 

Diff. 

1° 

Diff. 

1° 

Diff. 

Oo 

Diff. 

0° 

Diff. 

0° 

Diff; 

0 

1 

2 
3 
4 
5 

38    0.0 
39  17.3 
40  34.4 
4151.4 
43   8.1 
44  24.4 

77.3 

77.1 
77.0 
76.7 
76.3 

958.4 
1036.1 
11  11.5 
1144.4 
1215.0 
1243.1 

37.7 
35.4 
32.9 
30.6 
28.1 

1030.5 
952.9 
913.2 
831.3 
747.3 
7    1.2 

37.6 
39.7 
41.9 
44.0 
46.1 

40   5.6 
3851.6 
3737.6 
36  23.8 
3510.1 
33  56.8 

74.0 
74.0 
73.8 
73.7 
73.3 

9  2.1 
824.3 
748.8 
715.5 
644.6 
616.0 

37.8 
35.5 
33.3 
30.9 
28.6 

758.5 
834.8 
913.3 
954.0 
1036.9 
1121.8 

36.3 

38.5 
40.7 
42.9 
44.9 

75.9 

25.5 

48.0 

72.9 

26.2 

47.0 

6 

7 
8 
9 
10 

4540.3 
4655.5 
4810.2 
4924.1 
5037.1 

75.2 
74.7 
73.9 
73.0 

13  8.6 
1331.7 
1352.2 
1410.2 
1425.6 

23.1 
20.5 
18.0 
15.4 

613.2 
523.3 
431.5 
337.8 
242.5 

49.9 
51.8 
53.7 
55.3 

3243.9 
3131.4 
3019.5 
29   8.3 

2757.8 

72.5 
71.9 
71.2 
70.5 

5  49.8 
526.0 
5  4.7 
445.9 
429.5 

23.8 
21.3 
18.8 
16.4 

12    8.8 
12S7.7 
1348.6 
1441.3 
1535.8 

48.9 
50.9 
52.7 
54.5 

72.1 

12.8 

57.1 

69.7 

13.7 

56.2 

11 
12 
13 

14 
15 

5149.2 
53    0.4 
54  10.4 
55  19.3 
5626.9 

71.2 
70.0 
68.9 
67.6 

1438.4 
1448.5 
1456.1 
15  1.0 
15  3.3 

10.1 
7.6 
4.9 
2.3 

145.4 
046.8 

58.6 
60.2 
61.6 
63.0 

2648.1 
2539.3 
2431.5 
23  24.7 
2219.1 

68.8 
67.8 
66.8 
65.6 

415.8 
4  4.5 
355.8 
349.7 
346.1 

11.3 
8.7 
6.1 
3.6 

1632.0 
1729.9 
1829.3 
19  30.2 
20  32.5 

57.9 
59.4 
60.9 
62.3 

5946.6 
58  45.0 
5742  0 

66.2 

0.3 

64.3 

64.3 

0.9 

63.7 

16 
17 
18 
19 
20 

5733.1 
58  38.0 
5941.3 

64.9 
63.3 
61.7 
60.1 

15  3.0 
15  0.0 

1454.5 
1446.3 
1435.5 

3.0 
5.5 

8.2 
10.8 

5637.7 
5532.1 
5425.5 
5317.7 
52    8.9 

65.6 
66.6 

67.8 
68.8 

21  14.8 
2011.7 
1910.0 
18    9.8 
1711.0 

63.1 
61.7 
60.2 
58.8 

3  45.2 
346.8 
351.0 
357.8 
4  7.1 

1.6 
4.2 
6.8 
9.3 

2136.2 
2241.1 
2347.2 
2454.4 
26   2.0 

64.9 
66.1 
67.2 
68.2 

043.0 
143  1 

58.3 

13.3 

69.6 

57.1 

12.0 

69.1 

21 
22 
23 
24 
25 

241.4 
£38.0 
432.7 
525.5 
616.3 

56.6 
54.7 

(TO   Q 

50.8 

1422.2 
14  6.3 
1347.9 
1326.9 
13  3.5 

15.9 
18.4 
21.0 
23.4 

5059.3 
49  48.8 
48  37.6 
4725.7 
46  13.3 

70.5 
71.2 
71.9 
72.4 

1613.9 
1518.4 
1424.7 
1332.8 
1242.7 

55.5 
53.7 
51.9 
50.1 

419.1 
433.6 
450.6 
510.1 
532.1 

14.5 
17.0 
19.5 
22.0 

2711.7 
2821.6 
29  32.3 
3043.6 
31  55.5 

69.9 
70.7 
71.3 
71.9 

48.7 

25.8 

72.9 

48.2 

24.5 

72.3 

26 

27 
28 
29 
30 

7   5.0 
751.6 
836.1 
918.4 
958.4 

46.6 
44.5 
42.3 
40.0 

1237.7 
12  9.4 
1138.7 
11  5.8 
1030.5 

28.3 
30.7 
32.9 
35.3 

45   0.4 
4347.0 
42  33.4 
4119.6 
40    5.6 

73.4 
73.6 
73.8 
74.0 

1154.5 
11    8.3 
1024.1 
942.0 
9   2.1 

46.2 
44.2 
42.1 
39.9 

556.6 
623.5 

652.8 
724.5 

758.5 

26.9 
29.3 
31.7 
34.0 

33    7.8 
3420.5 
35  33.5 
36  46.7 
38    0.0 

72.7 
73.0 
73.2 
73.3 

1° 

1° 

0° 

Oo 

0° 

0° 

78      '  TABLE  LIII.     Reduction. 

Argument.     Supplement  of  Node  +  Moon's  Orbit  Longitude. 


Oa  Vis 

Diff. 

Is  VIIsDiff    IlaVIHs  Diff. 

Ills  IXs 

Diff. 

IVs  Xs 

Diff. 

VaXIs 

Diff.' 

0 

,      „ 

.      /, 

/     // 

/    ,/ 

,      „ 

,      // 

0 

1 

2 
3 
4 

5 

7    0.0 
6  45.6 
6  31.2 
6  16.9 
6     2.6 
5  48.4 

14.4 
14.4 
14.3 
14.3 
14.2 

1      3.0  '  " 

o  se.o  ••; 

0  49.5  J;J 
0  43.4  J-* 
0  37.8  f  J 
0  3S.70-1 

1     3.0 
1  10.4 
1   18.3 
1  26.5 
1  35.2 
1  44.2 

7.4 
7.9 

8.2 
8.7 
9.0 

7    0.0 

7  14.4 
7  28.8 
743.1 
7  57.4 
8  11.6 

14.4 
14.4 
14.3 
14.3 
14.2 

12  57.0 
13     4.0 
13  10.5 
13  16.6 
13  22.2 
13  27.3 

7.0 
6.5 
6.1 
5.6 
5.1 

12  57.0 
12  49.6 
12  41.7 
12  33.5 
12  24.8 
12   15.8 

7.4 
7.9 
8.21 
8.7; 

9.0 

14.1 

4.5 

9.5 

14.1 

4.5 

'9.5; 

6 

7 
8 
9 
10 

5  34.3 
5  20.3 
5     6.4 

4  52.6 
4  39.0 

14.0 
13.9 
13.8 
13.6 

0  28.2 
0  23.9  *3 
0  20.0  ,r 
0  16.8  ** 

0  14.1  ^  •' 

1  53.7 
2     8.5 
2  13.7 
2  24.2 
S  35.0 

9.8 
10.2 
10.5 
10.8 

8  25.7 
8  39.7 
8  53.6 
9    7.4 
9  21.0 

14.0 
13.9 
13.8 
13.6 

13  31.8 
13  36.1 
13  40.0 
13  43.2 
13  45.9 

4.3 
3.9 
3.2 

2.7 

12     6.3 
11   56.5 
11   46.3 
11   35.8 
11   25.0 

9.8 
10.2 
10.5 
10.8 

13.4 

2.3 

11.2 

13.4 

2.3 

11.2 

11 
12 
13 
14 
15 

4  25.6 
4  12.3 
3  59.3 
3  46.5 
3  33.9* 

13.3 
13.0 
12.8 
12.6 

0  11.8  .  „ 
0  10.1  :{•' 
0    8-8  i'? 
0    8.1  J' 

0     7.8  °'d 

2  46.2 
2  57.7 
3     9.5 
3  21.6 
3  33.9 

11.5 

11.8 
12.1 
12.3 

9  34.4 
9  47.7 
10    0.7 
10  13.5 
10  26.1 

13.3 
13.0 

12.8 
12.6 

13  48.2 
13  49.9 
13  51.2 
13  51.9 
13  52.2 

1.7 
1.3 
0.7 
0.3 

11    13.8 
11     2.3 

10  50.5 
10  38.4 
10  26.1 

11.5 
11.8 
12.1 
12.3 

12.3             ;0.3 

12.6 

12.3 

0.3 

12.6 

16 
17 

18 
19 

20 

3  21.6 
3     9.5 
2  57.7 
2  46.2 
2  35.0 

"»«     Si 

}{$  iai 

},  JO  11.8 
v   -,fl  14.1 

0.7 
1.3 
1.7 
2'.3 

3  46.5 
3  59.3 
4  12.3 
4  25.6 
4  39.0 

12.8  '}j}j!!-*'  12.1 
13.0  .„     go  H.8 

13.3    ,:i  11.5 

,n   A    11     10.  0     -  ,    ~ 

13  4;  11  25.0  1L2 

13  51.9 
13  51.2 
13  49.9 
13  48.2 
13  45.9 

0.7 
1.3 
1.7 
2.3 

10   13.5 
10     0.7 
9  47.7 
9  34.4 
9  21.0 

12.8 
13.0 
13.3 
13.4 

10.8 

2.7 

13.6  1 

10.8 

2.7 

13.6 

21 
22 
23 
24 
25 

2  24.2 
2  13.7 
2     3.5 
53.7 
44.2 

10.5 
10.2 
9.8 
9.5 

0  16.8 
0  20.0 
0  23.9 
0  28.2 
0  32.7 

3.2 
3.9 
4.3 
4.5 

4  52.6   ,„« 

5  6.4  *J; 

5  20.3   JJ'J 
5  34.3   JJJ 
5  48.4 

11  35.8 
11  46.3 
11  56.5 
12    6.3 
12  15.8 

10.5 
10.2 
9.8 
9.5 

13  43.2 
13  40.0 
13  36.1 
13  31.8 
13  27.3 

3.2 
3.9 
4.3 
4,5 

9     7.4 
8  53.6 
8  39.7 
8  25.7 
8   11.6 

13.8 
13.9 
14.0 
14.1 

9.0 

5.1 

14.2 

9.0 

5.1 

14.2 

2fi 
27 
28 
29 
30 

35.2 
28.5 
183 
10.4 
30 

8.7 
82 
7.9 
7.4 

0  37.8 
0  43.4 
0  49.5 
0  56.0 
1     3.0 

56 
6.1 
6.5 
7.0 

6     2.6 
6  16.9 
6  31.2 
6  45.6 
7     0.0 

1d«  12  24.8 
Jj  31'  12  33.5 
J™!  12  41.7 

14<4J12  57.0 

8.7 
8.2 
7.9 
7.4 

13  22.2 
13   16.6 
13   10.5 
13     4.0 
12  57.0 

5.6 
6.1 
6.5 
7.0 

7  57.4 
7  43.1 
7  28.8 
7   14.4 
7     0.0 

14.3 
14.3 
144 
M4 

TABLE  LIV.    Lunar  Nutation  in  Longitude. 
Argument.     Supplement  .of  the  Node. 


0* 

I* 

!!• 

Ills 

IV« 

Vs 

+ 

+ 

+ 

+ 

f 

_j_ 

o 

„ 

tt 

,, 

„ 

// 

// 

o 

0 

0.0 

8.5 

14.8 

17.3 

15.2 

8.8 

30 

2 

0.6 

9.0 

15.1 

17.2 

14.9 

8.1 

28  i 

4 

1.8 

9.4 

15.4 

17.2 

14.5 

7.7 

26 

6 

1.7 

10.0 

15.6 

17.2 

14.2 

7.2 

24 

8 

2.3 

10.4 

15.9 

17.2 

13.8 

6.5 

22 

10 

2.9 

10.9 

16.4 

17.1 

13.5 

6.1 

20 

12 

3.5 

11.4 

16.3 

17.0 

13.0 

5.4 

18 

14 

4.1 

11.8 

16.5 

16.9 

12.6 

4.8 

16 

16 

4.6 

12.2 

16.7 

16.7 

12.2 

4.3 

14 

18 

5.2 

12.6 

16.8 

16.5 

11.8 

3.7 

12 

20 

5.8 

13.1 

16.9 

16.4 

11.3 

3.0 

10 

22 

6.2 

13.4 

17.1 

16.2 

10.9 

2.4 

8 

24 

6.9 

13.8 

17.1 

15.9 

10.4 

1.8 

6 

26 

7.4 

14.1 

17.2 

15.7 

9.8 

1.3 

4 

28 

7.8 

14.5 

17.2 

15.4 

9.4 

0.6 

2 

30 

8.5 

14.8 

17.3 

15.2 

8.8 

0.0 

0 

XI« 

X* 

IX* 

VIII* 

VII* 

VI* 

TABLE  LV.  79 

Moon's  Distance  from  the  North  Pole  of  the  Ecliptic. 
Argument.     Supplement  of  Node+Moon's  Orbit  Longitude. 


III* 

IV* 

V* 

VI« 

VII* 

VIII* 

. 

84° 

85° 

Diff. 
for  10 

87° 

Diff. 
for  10 

89° 

Diff. 

for  10 

92° 

Diff. 

for  10 

94° 

Q  0 
30 
1  0 
30 
2  0 
30 

3916.0 
3916.7 
3918.8 
3922.4 
3927.3 
3933.7 

2042.7 
22   4.2 
23  27.0 
2451.0 
2616.2 
2742.6 

27.2 
27.6 
28.0 

28.4 
28.8 

1346.6 
16   6.9 
1827.8 
2049.5 
2311.8 
25  34.8 

46.8 
47.0 
47.2 
47.4 
47.7 

48   0.0 
5041.4 
5322.9 
56   4.3 

5845.7 
1270 

53.8 
53.8 
53.8 
53.8 
53.8 

/       // 

2213.4 
2433.1 
2652.2 
2910.2 
31  27.5 
3344.2 

4^.6 
46.4 
46.0 
45.8 
45.6 

/    // 
1517.3 
1637.7 
1756.8 
1914.6 
2031.3 
2146.7 

O      ' 

30  0 
30 
29  0 
30 

28  0 
30 

29  2 

47.9 

53.8 

45.3 

3  0 

30 
4  0 
30 
5  0 

3941.5 
39  50.6 
40    1.2 
4013.2 
4026.7 

2910.1 
3038.9 
32   8.8 
3339.9 
35  12.2 

29.6 
30.0 
30.4 
30.8 

2758.5 
30  22.8 
3247.7 
35  13.2 
3739.3 

48.1 
48.3 
48.5 
48.7 

4   8.3 
649.5 
930.6 
1211.6 
1452.5 

53.7 
53.7 
53.7 
53.6 

36    0.2 
3815.3 
40  29.7 
42  43.3 
4456.2 

45.0 
44.8 
44.5 
44.3 

23    0.8 
2413.7 
25  25.3 
26  35.7 
2744.8 

27  0 

do 

26  0 
30 
25  0 

31.1 

48.9 

53.6 

44.0 

30 
6  0 
30 
7  0 
30 

4041.5 
40  57.7 
41  15.4 
4134.4 
41  54.8 

3645.6 
3820.1 
39  55.8 
41  32.7 
43  10.6 

31.5 
31.9 
32.3 
32.6 

40    6.1 
42  33.4 
45    1.2 
4729.6 
4958.6 

49.1 
49.3 
49.5 
49.7 

1733.3 
20  14.0 
22  54.4 
2534.8 
28  14.9 

53.6 
53.5 
53.5 
53.4 

47   8.1 
49  19.4 
51  29.7 
5339.3 
5548.0 

43.8 
43.4 
43.2 
42.9 

28  52.6 
29  59.0 
31    4.3 
32   8.2 
33  10.9 

30 
24  0 
30 
23  0 
30 

33.0 

49.8 

53.3 

42.6 

8  0 
30 
9  0 
30 
10  0 

42  16.7 
4239.9 
43   4.6 
4330.6 
4358.1 

4449.7 
4629.9 
4811.2 
4953.5 
51  37.0 

33.4 
33.8 
34.1 
34.5 

5228.1 
54  58.2 
5728.7 
59  59.8 
~231  3 

50.0 
50.2 
50.4 
50.5 

3054.9 
3334.7 
3614.3 
3853.7 
41  32.8 

53.3 
53.2 
53.1 
53.0 

5755.8 
0   2.8 
2   8.9 
414.1 
618.4 

42.3 
42.0 
41.7 
41.5 

3412.2 
35  12.2 
36  10.9 
37   8.3 
38   4.4 

22  0 
30 
21  0 
30 
20  0 

34.9 

50.7 

53.0 

41.1 

30 
11  0 
30 
12  0 
30 

4426.9 
4457.1 
4528.8 
46    1.8 
4636.1 

5321.6 
55    7.1 
56  53.8 
5841.6 
"fj^fjlj 

35.2 
35.7 
35.9 
36.2 

5    3.3 
735.8 
10    8.8 
1242.1 
1516.0 

508  441L7 

5?'0  4650'4 
;}  ,  14928.7 

°  "I  52    6.8 
51<d  5444.6 

52.9 
52.8 
52.7 
52.6 

821.8 
1024.3 
1225.9 
1426.6 
1626.3 

40.8 
40.5 
40.2 
39.9 

3859.1 
3952.5 
4044,6 
41  35.3 
4224.7 

30 
19  0 
30 
18  0 
30 

36.6 

51.4 

52.5 

39.6 

13  0 
30 
14  0 
30 
15  0 

4711.9 
4749.0 
48  27.5 
49    7.4 
4948.7 

220.1 
4H.O 
6    2.9 
755.7 
949.6 

37.0 
37.3 
37.6 
38.0 

1750.2 
20  24.9 
2259.9 
2535.3 
2811.1 

51.6 

51.7 
51.8 
51.9 

5722.1 
59  59.3 
236.2 
512.7 
748.9 

52.4 
52.3 
52.2 
52.1 

1825.0 
2022.8 
2219.7 
24  15.5 
26  10.4 

39.3 
38.0 
38.6 
38.3 

43  12.7 
43  59.4 
4444.7 
4528.7 
4611.3 

17  0 
30 
16  0 
30 
15  0 

84°        86° 

88° 

91° 

93° 

94° 

II* 

I* 

0 

XI* 

•ix' 

IX- 

80  TABLE  LV. 

Moon's  Distance  from  the  North  Pole  of  the  Ecliptic. 
Argument.     Supplement  of  Node+tyfoon's  Orbit  Longitude. 


III* 

IV* 

V* 

VI* 

VII* 

VIII* 

84° 

86° 

Diff. 

for  10 

88° 

Dirt, 
for  10 

91° 

Diff. 

for  10 

93° 

Ditf: 

for  10 

94° 

O    ' 

/    /' 

/     /> 

,    // 

,   // 

/    // 

/  // 

O       ' 

15  0  49  48.7    9  49.6 
30  50  31.3  ill  44.5 
16  051  15.3  1  13  40.3 
3052    0.6|15?72 
17  052  47.3i  173o.O 
30  53  35.3  19  33.7 

38.3 
38.6 
39.0 
39.3 
39.6 

28  11.1 
30  47.3 
33  23.8 
36    0.7 
38  37.9 
41  15.4 

52.1 
52.2 
52.3 

52.4 
52.5 

748.9 
10  24.7 
13    0.1 
1535.1 
18    9.8 
20  44.0 

51.9 
51.8 
51.7 
51.6 
51.4 

26  10.4 
28    4.3 
29  57.1 
31  49.0 
33  39.9 
35  29.7 

38.0 
37.6 
37.3 
37.0 
36.6 

46  11.3 
46  52.6 
47  32.5 
48  11.0 
4848.1 
49  23.9 

15  0 
30 
14  0 
30 
13  0 
30 

39.9 

52.6 

51.3 

36.2 

18  0  54  24.7  21  33.4 
3055  15.4  2334.1 
19  056    7.5  2535.7 
3057    0.92738.2 
20  0  57  55.6  29  41.6 

40.2 
40.5 
40.8 
41.1 

43  53.2 
4631.3 
49    9.6 
51  48.3 
54  27.2 

52.7 
52.8 
52.9 
53.0 

23  17.9 
25  51.2 

28  24.2 
30  56.7 
33  28.7 

51.1 
51.0 
50.8 
50.7 

37  18.4 
39    6.2 
40  52.9 
42  38.4 
44  23.0 

35.9 
35.6 
35.2 
34.9 

49  58.2 
5031.2 
51  2.9 
51  33.1 
52  1.9 

12  0 
30 
11  0 
30 
10  0 

41.4 

53.0 

50.5 

34.5 

305851.7  31  45.9 
21  0  59  49.1:3351.1 
30    047.83557.2 
22  0'   1  47.8  j  38    4.2 
30    249.1  40  12.0 

41.7 
42.0 
42.3 
42.6 

42.9 

57    6.3 
59  45.7 
T25^3 
5    5.1 
745.1 

53.1 
53.2 
53.3 
53.3 

53.4 

36    0.2 
3831.3 
41    1.8 
4331.9 
46    1.4 

50.4 
50.2 
50.0 
49.8 

49.7 

46    6.5 
47  48.8 
49  30.1 
51  10.3 
52  49.4 

34.1 
33.8 
33.4 
33.0 
32.6 

52  29.4 
52  55.4 
5320.1 
53  43.3 
54  5.2 

30 
9  0 
30 
8  0 
30 

23  0 
30 
24  0 
30 

9,5  0 

351.8  4220.7 
4  55.7  44  30.3 
6    1.0!4640.6 
7    7.4J4S51.9 
815.251    3.8 

43.2 
43.4 
43.6 
44.0 

10  25.2 
13    5.6 
15  46.0 

18  26.7 
21    7.5 

53.5 
5-3.5 
53.6 
53.6 

48  30.4 
50  58.8 
53  26.6 
55  53.9 
58  20.7 

49.5 
49.3 
49.1 

48.9 

5427.3 
56    4.2 
57  39.9 
59  14.4 
0  47  8 

32.3 
31.9 
31.5 
31.1 

5425.6 
54  44.6 
55  2.3 
55  18.5 
55  33.3 

7  0 
30 
6  0 
30 
5  0 

443 

53.6 

48.7 

30.8 

30 

26  0 
30 
27  0 
30 

924.3 
10  34.7 
11  46.3 
12  59.2 
14  13.3 

53  16.7 
55  30.3 

57  44.7 
59  59.8 

44.5 
44.8 
45.0 
45.3 

23  48.4 
26  29.4 
29  10.5 
31  51.7 
34  33.0 

53.7 
53.7 
53.7 
53.7 

046.8 
3  12  3 

537.2 
8    1.5 
1025.2 

48.5 

48.3 
48.2 
47.9 

220.1 
351.2 
521.1 
649.9 
8  17.4 

30.4 
30.0 
29.6 

29.2 

55  46.8 
55  58.8 
56  9.4 
56  18.5 
56  26.3 

30 
4  0 
30 
3  0 
30 

2158 

45.6 

53.8 

47.7 

28.8 

28  0  15  28.7 
30  16  45.4 
29  0  18    3.2 
30  19  22.3 
30  0  20  42.7 

432.5 
649.8 
9    7.8 
11  26.9 
1346.6 

45.8 
46.0 
46.4 
46.6 

37  14.3 
39  55.7 
42  37.1 
45  18.6 
48    0.0 

53.8 
53.8 
53.8 
53.8 

1248.2 
15  10.5 
17  32.2 
1953.1 
22  13.4 

47.4 
47.2 
47.0 
46.7 

943.8 
11.  9.0 
12330 
13  55.5 
15  17.3 

28.4 
280 
27.6 
27.2 

56  32.7 
56  37.6 
5641.2 
56  43.3 
56  44.0 

2  0 

30 
1  0 
30 
0  0 

85°        |87° 

89° 

j  92° 

94° 

94° 

II*    1     I* 

O 

I    XI* 

X* 

IX* 

TABLE  LVI. 

Equation  II  of  the  Moon's  Polar  Distance. 
Argument  II,  corrected. 


81 


III* 

diff. 

IV* 

diff. 

V* 

diff. 

VI* 

diff. 

VII* 

diff. 

VIII* 

diff. 

0 

„ 

/  // 

,   ,, 

/     // 

/    „ 

/  // 

o 

0 

1 

2 
3 
4 
5 

0  13.8 
0  13.9 
0  14.1 
0  14.5 
0  15.1 
0  15.8 

0.1 
0.2 
0.4 
0.6 
0.7 

1  24.4 
1  29.0 
1  33.8 
1  38.7 
143.8 
149.0 

4.6 

4.8 
4.9 
5.1 
5.2 

436.9 
444.9 
4530 
5    1.1 
5    9.3 
517.6 

8.0 

8.1 
8.1 
8.2 
8.3 

9    0.0 
9    9.2 
9  18.4 
927.5 
936.7 
9  45:9 

9.2 
9.2 
9.1 
9.2 
9.2 

13  23.1 
1331.0 
13  38.8 
1346.6 
13  54.2 
14    1.8 

7.9 

7.8 
7.8 
7.6 
7.6 

1635.6 
16  40.2 
16  44.6 
16  48.9 
16  53.0 
16  56.9 

4.6 
4.4 
4.3 
4.1 
3.9 

30 
29 
28 
27 
26 
25 

0.9 

5.3 

8.4 

9.1 

7.5 

3.8 

6 

7 
8 
9 
10 

0  16.7 
0  17.7 
0  18.9 
020.3 
021.8 

1.0 
1.2 
1.4 
1.5 

1  54.3 
1  59.8 
2  5.4 
211.1 
2  16.9 

55 
5.6 
5.7 

5.8 

526.0 
534.4 
542.9 
551.4 
6    0.0 

8.4 
8.5 
8.5 
8.6 

955.0 
10    4.1 
10  13.2 
10  22.3 
1031.4 

9.1 
9.1 
9.1 
9.1 

14    9.3 
1416.7 
14  24.0 
1431.2 
14  38.2 

7.4 
7.3 
7.2 
7.0 

17    0.7 
17    4.4 
17    7.9 
1711.3 
17  14.5 

3.7 
3.5 

3.4 
3.2 

24 
23 
22 
21 
20 

1.7 

6.0 

8.7 

9.0 

|7.0 

3.0 

11 

12 
13 
14 
15 

023.5 
025.3 
027.3 

029.4 
031.7 

1.8 
2.0 
2.1 
2.3 

222.9 
2  29.0 
235.2 

241.5 
247.9 

6.1 
6.2 
6.3 
6.4 

6    8.7 
6  17.4 
626.2 
635.0 
643.8 

8.7 
8.8 
8.8 
8.8 

10  40.4 
10  49.4 
10  58.4 
11    7.3 
11  16.2 

9.0 
9.0 
8.9 
8.9 

1445.2 
1,4  52.1 
14  58.9 
15    5.5 
15  12.1 

6.9 
6.8 
6.6 
6.6 

17  17.5 
17  20.4 
17  23.2 
17  25.8 
17  28.3 

2.9 
2.8 
2.6 
2.5 

19 
18 
17 
16 
15 

2.5 

6.6 

8.9 

8.8 

6.4 

2.3 

16 
17 
18 
19 
20 

034.2 
036.8 
039.6 
042.5 
0  45.5 

2.6 
2.8 
2.9 
3.0 

254.5 
3  1.1 
3  7.9 
3  14.8 
321.8 

6.6 
6.8* 
6.9 
7.0 

652.7 
7    1.6 
7  10.6 
719.6 

728.6 

8.9 
9.0 
9.0 
9.0 

11  25.0 
11  33.8 
11  42.6 
11  51.3 
12    0.0 

8.8 
8.8 
8.7 
8.7 

15  18.5 
15  24.8 
1531.0 
15  37.1 
1543.1 

6.3 
6.2 
6.1 
6.0 

17  30.6 
17  32.7 
17  34.7 
17  36.5 
1738.2 

2.1 
2.0 
1.8 
1.7 

14 
13 
12 
11 
10 

3.2 

7.0 

9.1 

8.6 

5.8 

Ii5 

21 
22 
23 
24 
25 

048.7 
052.1 
0  55.6 
0  59.3 
1    3.1 

343288 

353360 
37  3  43.3 

j!  3  50.7 
d'8  3  58.2 

7.2 
7.3 

7.4 
7.5 

737.7 
746.8 
7  55.9 
8    5.0 
8  14.1 

9.1 
9.1 
9.1 
9.1 

12    8.6 
12  17.1 
12  25.6 
12  34.0 

12  42.4 

8.5 
8.5 
8.4 
8.4 

15  48.9 
15  54.6 
16    0.2 
16    5.7 
16  11.0 

5.7 
5.6 
5.5 
5.3 

17  39.7 
1741.1 
1742.3 
1743.3 

17  44.2 

1.4 
1.2 
1.0 
0.9 

9 

8 
7 
6 
5 

3.9 

7.6 

9.2 

8.3 

5.2 

0.7 

26 

27 
2S 
29 
30 

1    7.0 
1  11.1 
1  15.4 
1  19.8 
1  24.4 

4      4    5.8 

I  I'i  4  13  4 

I;?  481.3 

46  4  29-° 
4  b  4  36.9 

7.6 

7.8 
7.8 
7.9 

,8  23.3 
8  32.5 
841.6 
8  50.8 
9    0.0 

9  2  12  50.7 
of  11258.9 
9  o  13    7.0 
9211315'1 

;  13  23.1 

8.2 
8.1 
8.1 
8.0 

16  16.2 
1621.3 
16  26.2 
16  31.9 
1635.6 

51  *744'9 
AQ  1745.5 
?  si  17  45.9 
~  1746.1 
4'6  il7  46.2 

0.6 
0.4 
0.2 
0.1 

4 
3 

2 
1 
0 

II* 

I* 

0* 

I    XI* 

X*             j  IX* 

TABLE  LVII. 

Equation  III  of  Moon's  Polar  Distance. 
Argument.     Moon's  True  Longitude. 


III* 

IV* 

V* 

VI* 

VII* 

VIII* 

0 

0 

16.0 

14.9 

12.0 

8.0 

4.0 

1.1 

0 

30 

6 

16.0 

14.5 

11.3 

7.2 

33 

0.7 

24 

12 

15.8 

13.9 

10.5 

6.3 

2.6 

0.4 

18 

18 

15.6 

13.4 

9.7 

5.5 

2.1 

0.2 

12 

24 

15.3 

12.7 

8.8 

4.7 

1.5 

0.0 

6 

30 

14.9 

12.0 

8.0 

4.0 

1.1 

0.0 

0 

II* 

I* 

O* 

XI* 

X* 

IX* 

:  "1 

F2TABLE  LVI1I. 

To  convert  Degrees 
and  Minutes  into 
Decimal  Parts. 


TABLE  LIX. 

Equations  of  Moon's  Polar  Distance. 
Arguments,  Arg.  20  of  Long. ;  V  to  IX 

corrected;  X  not  corrected;  and  XI 

and  XII  corrected. 


Deg.  Dec 

&Mm.  parts. 

0   ' 

1  5 

003 

1  26 

4 

148 

5 

2  10 

6 

231 

7 

253 

8 

3  14 

9 

336 

10 

358 

11 

4  i'J 

12 

441 

13 

5  2 

14 

524 

15 

546 

16 

6  7 

17 

629 

18 

650 

19 

7  12 

20 

734 

21 

755 

22 

8  17 

23 

838 

24 

9  0 

25 

922 

26 

943 

27 

10  5 

28 

1026 

29 

1043 

30 

11  10  31 

11  31 

32 

1153 

33 

12  14 

34 

1236 

3f> 

1258 

36 

1319 

37 

1341 

38 

14  2 

39 

1424 

40 

1446 

41 

15  7 

42 

1529 

43 

1550 

44 

16  12 

45 

1634 

46 

1655 

47 

1717 

48 

1738 

49 

18  0 

50 

1822!  51 

1843  52 

19  5  53 

Arg 

20 

V 

VI 

VII 

VIE 

IX 

X 

XI 

Arg 

Arg 

XII 

Arg. 

250 

0.3 

55.9 

6.1 

2.6 

25.1 

3.0 

0.7 

0.9 

250 

0 

4.0 

500 

260 

0.3 

55.8 

6.2 

2.725.1 

3.1  0.7 

0.9 

240 

10 

3.7 

510 

270 

0.4 

55.7 

6.3 

2.8 

25.0 

3.2!  0.8 

1.0 

230 

20 

3.4 

520 

280 

0.6 

55.4 

6.5 

3.0  24.9 

3.5 

1.0 

1.0 

220 

30 

3.1 

530 

290 

0.8 

55.1 

6.9 

3.324.8 

3.8  1.2 

1.1 

210 

40 

2.8 

540 

300 

1.0 

54.6 

7.3 

3.724.7 

4.3  1.5 

1.2 

200 

50 

2.5 

550 

310 

1.3 

54.1 

7.8 

4.2  24.4 

4.9  1.8 

1.3 

190 

60 

2.3 

5GO 

320 

1.7 

53.4 

8.4 

4.7,24.1 

5.6  2.2 

1.4 

180 

70 

2.1 

570 

330 

2.1 

52.7 

9.1 

5.423.8 

6.4  2.7 

1.5 

170 

80 

1.9 

580 

340 

2.6 

51.9 

9.8 

6.1 

23.5 

7.2!  3.2 

1.7 

160 

90 

1.7 

590 

1 

350 

3.1 

51.0 

10.7 

6.9  23.2 

8.2 

3.8 

1.9 

150 

100 

1.6 

600 

360 

3.7 

50.0 

11.6 

7.7,22.8 

9.2 

4.4 

2.1 

140 

110 

1.5 

610 

370 

4.3 

48.9 

12.6 

8.7J22.4 

10.3 

5.1 

2.3 

130 

120 

1.5 

620 

380 

4.9 

17.7 

13.6 

9.7:21.9 

11.5 

5.8 

2.5 

120 

130 

1.5 

630 

390 

5.6 

46.5 

14.8 

10.721.4 

12.8 

6.6 

2.8 

110 

140 

1.5 

640 

400 

6.4 

45.2 

16.0 

11.820.9 

14.1 

7.4 

3.0 

100 

150 

1.6 

650 

410 

7.1 

43.9 

17.2 

13.020.4 

15.5 

8.3 

3.3 

90 

160 

1.7 

660 

420 

7.9 

42.5 

18.5 

14.2 

19.9 

17.0 

9.1 

3.5 

80 

170 

1.9 

670 

430 

8.8 

41.0 

19.8 

15.5  19.3 

18.5 

10.1 

3.8 

70 

180 

2.1 

680 

440 

9.6 

39.5 

21.2 

16.8 

18.7 

20.1 

11.0 

4.1 

60 

190 

2.3 

690 

450 

10.5 

38.0 

22.6 

18.1 

18.1 

21.7 

12.0 

4.4 

50 

200 

2.5 

700 

460 

11.3 

36.4 

24.1 

19.4 

17.5 

23.3 

12.9 

4.7 

40 

210 

2.8 

710 

470 

12.2 

34.9 

25.5 

20.8 

16.9 

24.9 

13.9 

5.0 

30 

220 

3.1 

720 

480 

13.2 

33.2 

27.0 

22.2 

16.3 

26.6 

15.0 

5.4 

20 

230 

3.4 

730 

490 

14.1 

31.6 

28.5 

23.6 

15.6 

28.3 

16.0 

5.7 

10 

240 

3.7 

740 

500 

150 

30.0 

30.0 

250 

15.0 

30.0 

17.0 

6.0 

0 

250 

4.0 

750 

510  159 

284 

31.5 

26.4  14.4 

31.7 

18.0 

6.3 

990 

260 

4.3 

760 

520 

16.8 

26.8 

33.0 

27.8 

13.7 

33.4 

19.0 

6.6 

980 

270 

4.6 

770 

530 

17.8 

25.1 

34.5 

29.2 

13.1 

35.1 

20.1 

7.0 

970 

280 

4.9 

780 

540 

18.7 

23.6 

35.9 

30^6 

12.5 

36.7 

21.1 

7.3 

960 

290 

5.2 

790 

550 

19.5 

22.0 

37.4 

31.9 

11.9 

38.3 

22.0 

7.6 

950 

300 

5.5 

800 

5GO 

20.4 

20.5 

38.8  33.2  11.3 

39.9 

23.0 

7.9 

940 

310 

5.7 

810 

570 

21.2 

19.0 

40.2  34.5 

10.7 

41.5 

23.9 

8.2 

930 

320 

5.9 

820 

580 

22.1 

17.5 

41.5  35.8 

10.1 

43.0 

24.9 

8.5 

920 

330 

6.1 

830 

590 

22.9 

16.1 

42.8  37.0 

9.6 

44.5 

25.7 

8.7 

910 

340 

6.3 

840 

600 

23.6 

14.8 

44.0  38.2 

9.1 

45.9 

26.6 

9.0 

900 

350 

6.4 

850 

610 

24.4 

13.5 

45.2,39.3 

8.6 

47.2 

27.4 

9.2 

890 

360 

6.5 

860 

620 

25.1 

12.3 

46.440.3 

8.1 

48.5 

28.2 

9.5 

880 

370 

6.5 

870 

630 

25.7 

11.1 

47.4'41.3 

7.6 

49.7 

28.9 

9.7 

870 

380 

6.5 

880 

640 

26.3 

10.0 

48.4  42.3 

7.2 

50.8 

29.6 

9.9 

860 

390 

6.5 

890 

650 

26.9 

9.0 

49.343.1 

6.8 

51.8 

30.2 

10.1 

850 

400 

6.4 

900 

660  27.4 

8.1 

50.2  43.9 

6.5 

52.8 

30.8 

10.3 

840 

410 

6.3 

910 

670  27.9 

7.3 

50.9144.6 

6.2 

53.6 

31.3 

10.5 

830 

420 

6.1 

920 

680 

28.3 

6.6 

51.645.3 

5.9 

54.4 

31.8 

10.6 

820 

430 

5.9 

930 

690 

28.7 

5.9 

52.2  45.8 

5.6 

55.1 

32.2 

10.7 

810 

440 

5.7 

940 

700 

29.0 

5.4 

52.7  46.3 

5.3 

55.7  32.5 

10.8 

800 

450 

5.5 

950 

710 

29.2 

4.9 

53.1  46.7 

5.2 

56.2,32.8 

10.9 

790 

460 

5.2 

960 

720 

29.4 

4.6 

53.5  47.0 

5.1 

56.5  33.0 

11.0  780 

470 

4.9 

970 

730  29.6 

4.3 

53.7  47.2 

5.0 

56.8  33.2 

11.0  770 

480 

4.6 

980 

740  99.7 

4.2 

53.8  47.3 

4.9 

56.933.3 

11.  1 

760 

490 

4.3 

990 

750  J29.7 

4.1 

53.947.4 

4.9 

57.033.3  11.1 

750 

500 

4.0  1000 

Constant  10" 


TABLE  LX.  TABLE  LXI.          S3 

•Small  Equations  of  Moon's  Parallax.       Moon's  Equatorial  Parallax. 

\rgs.,  1,  2,  4,  5,  6,  8, 9, 12, 13,  of  Long.         Argument.  Arg.  of  Evection. 


A. 

I 

2 

4 

5 

6 

8 

9 

12 

13 

A. 

0 

0.0 

1.6 

0.6 

1.6 

1.9 

0.0 

3.0 

.4 

2.0 

100 

30.0 

1.6 

0.6 

1.6 

1.9 

0.0 

3.5 

.4 

2.0 

97 

6 

0.0 

1.5 

0.6 

1.5 

1.8 

0.0 

3.1 

.4 

1.9 

94 

9 

0.1 

1.5 

O.G 

.5 

1.8 

0.1 

2.6 

.3 

1.8 

91 

120.1 

1.4 

0.5 

.4 

.7 

0.2 

1.9 

.2 

1.7 

88 

150.1 

1.3 

0.5 

.3 

.6 

0.2 

1.3 

.1 

1.6 

85 

180.2 

1.1 

0.4 

.1 

.4 

0.3 

0.7 

1.0 

1.4 

82 

21 

0.3 

1.0 

0.4 

.0 

.3 

0.5 

0.2 

0.9 

1.2 

79 

240.4 

0.9 

0.3 

0.9 

.2 

0.6 

0.0 

0.7 

1.0 

76 

270.5 

0.7 

0.3 

0.7 

1.0 

0.7 

0.1 

0.6 

0.9 

73 

30  0.5 

0.6  0.2 

0.6 

0.9 

0.8 

0.4 

0.5 

0.7 

70 

330.6 

0.4  0.2 

0.4 

0.7 

0.9 

0.8 

0.4 

0.5 

67 

360.7 

0.3 

0.1 

0.3 

0.6 

1.0 

1.5 

0.3 

0.4 

64 

390.7 

0.2 

0.1 

0.2 

0.5 

1.1 

2.1  0.2 

0.2 

61 

420.8 

0.1 

0.0 

0.1 

0.4 

l.l 

2.8  0.1 

0.1 

58 

45 

0.8 

0.0 

0.0 

0.0 

0.3 

1.2 

3.2 

0.0 

0.0 

55 

480.8 
500.S 

0.0 
0.0 

0.0 
0.0 

0.0 
0.0 

0.3 
0.3 

1.2 
1.2 

3.5  0.0 

3.6)0.0 

0.0 
0.0 

52 
50 

Constant    7" 

The  first  two  figures  only  of  the  Arguments 

are  taken. 

o 

I* 

II" 

III* 

IV* 

V* 

0 

0 

120.8 

1  15.6 

1    1.6 

42.6 

24.1 

10.8 

0 

30 

1 

1  20.8J1  15.2 

1    0.9 

41.9 

23.6 

10.5 

29 

2 

1  20.8 

1  14.9 

1    0.3 

41.3 

23.0 

10.2 

28 

3 

1  20.7  1  14.5 

59.7 

40.6 

22.5 

9.9 

27 

4 

1  20.7     14.2 

59.2 

40.0 

21.9 

9.6 

26 

5 

1  20.6      13.8 

58.6 

39.4 

21.4 

9.4 

25 

6 

1  20.6      13.4 

57.9 

38.7 

20.9 

9.1 

24 

7 

1  20.5;     13.0 

57.3 

38.1 

20.4 

8.8 

23J 

8 

1  20.4!     12.6 

56.7 

37.4 

19.9 

8.6 

221 

9 

1  20.3      12.2 

56.1 

36.8 

19.4 

8.4 

21 

10 

1  20.2;     11.7 

55.5 

36.1 

18.9 

8.2 

20i 

11 

120.1      11.3 

54.9 

35.5 

18.4 

8.0 

19 

12 

19.9      10.8 

54.2 

34.9 

17.9 

7.8 

18 

13 

19.8      10.4 

53.6 

34.2 

17.5 

7.6 

17 

14 

19.6        9.9 

53.0 

33.6 

17.0 

7.4 

16 

15 

19.5  ,       9.4 

52.3 

33.0 

16.6 

7.2 

15 

16 

19.3       9.0 

51.7 

32.4 

16.1 

7.1 

14 

17 

19.1        8.5 

51.1 

31.7 

15.7 

6.9 

13 

18 

18.9       8.0 

50.4 

31.1 

15.2 

6.8 

12 

19 

18.7       7.5 

49.8 

30.5 

14.8 

6.7 

11 

20 

18.4       7.0 

49.1 

29.9 

14.4 

6.5 

10 

21 

18.2       6.5 

48.5 

29.3 

14.0 

6.4 

9 

22 

18.0       5.9 

47.8 

28.7 

13.6 

6.3 

8 

23 

17.7,      5.4 

47.2 

28.1 

13.2 

6.3 

7 

24 

17.4      4.8 

46.5 

27.5 

12.9 

6.2 

6 

25 

17.1 

4.3 

45.9 

26.9 

12.5 

6.1 

5 

26 

16.9 

3.8 

45.2 

26.3 

12.1 

6.1 

4 

27 

16.6 

3.2 

44.6 

25.8 

11.8 

6.1 

3 

28 

16.2 

2.6 

43.9 

25.2 

11.5 

6.0 

2 

29 

15.9 

2.1 

43.3124.7 

11.11  6.0 

1 

30 

15.6 

1.5 

42.6!24.l|l0.8l  6.0 

°! 

XI* 

X* 

IX*  IvTIlJviI* 

VI* 

84 


TABLE  LXII. 
Moon's  Equatorial  Parallax. 

Argument.      Anomaly. 


0* 

diff 

I* 

diff     II* 

diff 

III* 

diff 

IV* 

diff     V«     diff 

0 

,    „ 

/    „ 

,    „ 

/    „ 

/    // 

/    „ 

o 

0 
1 
2 
3 
4 
5 

5857.7 
58  57.  7  1 
5857.6! 
58  57.4! 
5857.1 
5856.8 

0.0 
0.1 
0.2 
0.3 
0.3 

58  27-.0 
58  25.0 
58  23.0 
58  20.9 
5818.7 
58  16.5 

2.0 
2.0 

2.1 

2.2 
2.2 

57   7.9 
57   4.8 
57    1.6 
5658.4 
5655.2 
56  52.0 

3.1 
3.2 
3.2 
3.2 
3.2 

5529.8 
5526.6 
5523.4 
5520.2 
55  17.0 
5513.8 

3.2 
3.2 
3.2 
3.2 
3.2 

54   1.9 
53  59.4 
53  56.9 
53  54.5 
5352.1 
5349.7 

2.5 
2.5 

2.4 
2.4 
2.4 

53   3.2 
53    1.8 
53   0.5 
52  59.3 
5258.1 
52  57.0 

1.4 
1.3 
1.2 
1.2 
1.1 

30 

29 
28 
27 
26 
25 

0.4 

2.2 

3.2 

3.2 

2.3 

1  2 

6 

7 
8 
9 
10 

58  56.4 
5856.0 
5855.4 
58  54.8 
58  54.2 

0.4 
0.6 
0.6 
0.6 

58  14.3 
58  12.0 
58    9.6 
58    7.2 
58    4.8 

2.3 

2.4 
2.4 
2.4 

56  48.8 
5645.5 
56  42.3 
56  39.0 
5635.7 

3.3 
3.2 
3.3 
3.3 

5510.6 
55    7.5 
55   4.4 
55    1.3 
54  58.2 

3.1 
3.1 
3.1 
3.1 

53  47.4 
5345.1 
5342.9 
5340.6 
53  38.5 

2.3 

2.2 
2.3 
2.1 

52  55.8 
52  54.8 
5253.8 
52  52.8 
5251.9 

1.0 
1.0 
1.0 
0.9 

24 
23 
22 
21 
20 

0.8 

2.5 

3.3 

3.1 

2.2 

0.9 

11 
12 
13 
14 
15 

58  53.4 
5852.6 
5851.8 
58  50.8 
58  49.8 

0.8 
0.8 
1.0 
1.0 

58    2325 
5759.8*'° 

5757.2™ 
5754.6J!! 
5751.9^' 

5632.4 
5629.1 
56  25.8 
56  22.5 
56  19.2 

3.3 
3.3 
3.3 
3.3 

5455.1 

5452.1 
5449.1 
5446.1 
5443.1 

3.0 
3.0 
3.0 
3.0 

5336.3 
53  34.2 
5332.1 
5330.1 
5328.1 

2.1 
2.1 
2.0 
2.0 

5251.0 
5250.1 
5249.3 
5248.6 
52  47.9 

0.9 
0.8 
0.7 
0.7 

19 
18 
17 
16 
15 

1.1 

2.7 

3.3 

2.9 

1.9 

0.7 

j 

16 
17 
18 
19 
20 

5848.7 
5847.6 
58  46.4 
5845.1 
5843.8 

1.1 
1.2 
1.3 
1.3 

5749.2 
57  46.4  2-8 
5743.7C 
5740.8~Q 
57  38.0  3'8 

56  15.9 
5612.6 
56    9.3 
56    6.0 
56   2.7 

3.3 
3.3 
3.3 
3.3 

5440.2 
5437.3 
5434.4 
5431.5 
5428.7 

2.9 
2.9 
2.9 
2.8 

5326.2 
53  24.3 
5322.4 
5320.6 
53  18.8 

1.9 
1.9 
1.8 
1.8 

5247.2 
5246.6 
5246.0 
5245.5 
5245.0 

0.6 
0.6 
0.5 
0.5 

14 
13 
12 
11 
10 

1.4 

2.9 

3.4 

2.8 

1.8 

0.4 

21 
22 
23 
24 
25 

58  42.4 
5840.9 
58  39.4 
58  37.8 
58  36.2 

1.5 
1.5 
1.6 
1.6 

57351 
5732.2 
5729.3 
5726.3 
5723.3 

2.9 
2.9 
3.0 
3.0 

55  59.3 
5556.0 
5552.7 
5549.4 
5546.1 

3.3 
3.3 
3.3 
3.3 

5425.9 
5423.1 
5420.3 
5417.6 
5414.9 

2.8 
2.8 
2.7 
2.7 

53  17.0 
53  15.3 
53  13.7 
53  12.0 
53  10.4 

1.7 
1.6 
1.7 
1.6 

5244.6 
5244.2 
5243.8 
5243.5 
5243.3 

0.4 
0.4 
0.3 
0.2 

9 
8 
7 
6 
5 

1.8 

3.0 

3.3 

2.7 

1.5 

0.2 

26 
27 
28 
29 
30 

58  34.4 
58  32.7 
58  30.9 
5829.0 
58  27.0 

1.7 
1.8 
1.9 
20 

5720.2 
57  17.2 
5714.1 
5711.0 
57   7.9 

3.0 
3.1 
3.1 
3.1 

5542.8 
5539.6 
5536.4 
5533.1 
5529.8 

3.2 
3.2 
3.3 
3.3 

54  12.2 
54   9.6 
54   7.0 
54  4.4 
54   1.9 

2.6 
2.6 
2.6 
2.5 

53   8.9 
53   7.4 
53   5.9 
53   4.5 
53   3.2 

,  ,  5243.1 

15242.9 

\l  5242.8 

J'J  '5242.7 
d  5242.7 

0.2 
0.1 
0.1 
0.0 

4 
3 
2 
1 
0 

XI* 

X* 

IX* 

VIII* 

VII* 

VI* 

TABLE   LXIII. 


85 


Moorfs  Equatorial  Parallax. 
Argument.     Argument  of  the  Variation, 


0 

Is 

II* 

III* 

IV* 

V« 

o 

„ 

„ 

„ 

,/ 

,/ 

// 

o 

0 

55.6 

42.3 

16.0 

3.7 

17.6 

44.0 

30 

1 

55.6 

41.5 

15.3 

3.8 

18.5 

44.8 

29 

2 

55.5 

40.7 

14.5 

3.8 

19.3 

45.6 

28 

3 

55.5 

39.8 

13.8 

3.9 

20.1 

46.3 

27 

4 

55.3 

39.0 

13.1 

4.1 

21.0 

47.0 

26 

5 

55.2 

38.1 

12.4 

4.3 

21.9 

47.7 

25 

6 

55.0 

37.2 

11.7 

4.5 

22.7 

48.4 

24 

7 

54.8 

36.3 

11.1 

4.7 

23.6 

49.1 

23 

8 

54.6 

35.5 

10.4 

5.0 

24.5 

49.7 

22 

9 

54.3 

34.6 

9.8 

5.3 

25.4 

50.3 

21 

10 

54.0 

33.7 

9.2 

5.6 

26.3 

50.9 

20 

11 

53.7 

32.7 

8.7 

6.0 

27.2 

51.5 

19 

12 

53.3 

31.8 

8.2 

6.3 

28.2 

52.1 

18 

13 

52.9 

30.9 

7.7 

6.8 

29.1 

52.6 

17 

14 

52.5 

30.0  - 

7.2 

7.2 

30.0 

53.1 

16 

15 

52.0 

29.1 

6.7 

7.7 

30.9 

53.5 

15 

16 

51.5 

28.2 

6.3 

8.2 

31.8 

54.0 

14 

17 

51.0 

27.2 

5.9 

8.7 

32.8 

54.4 

13 

18 

50.5 

26.3 

5.6 

9.3 

33.7 

54.8 

12 

19 

49.9 

25.4 

5.3 

9.8 

34.6 

55.1 

11 

20 

49.4 

24.5 

5.0 

10.5 

35.5 

55.4 

10 

21 

48.8 

23.6 

4.7 

11.1 

36.4 

55.7 

9 

22 

48.1 

22.7 

4.5 

11.7 

37.3 

56.0 

8 

23 

47.4 

21.9 

4.3 

12.4 

38.2 

56.2 

7 

24 

46.8 

21.0 

4.1 

13.1 

39.0 

56.4 

6 

25 

46.1 

20.1 

3.9 

13.8 

39.9 

56.6 

5 

26 

45.4 

19.3 

3.8 

14.5 

40.8 

56.8 

4 

27 

44.6 

18.5 

3.7 

15.3 

41.6 

56.9 

3 

28 

43.9 

17.6 

3.7 

16.1 

42.4 

56.9 

2 

29 

43.1 

16.8 

3.7 

16.8 

43.2 

57.0 

1 

30 

42.3 

16.0 

3.7 

17.6 

44.0 

67.0 

0 

55 

X* 

IX* 

VIII* 

VII* 

VI* 

S6        TABLE  LXIV. 


TABLE  LXV. 


Reduction  of  the  Parallax, 
and  also  of  the  Latitude. 

Argument.     Latitude. 


Moon's  Semi-diameter. 
Argument.     Equatorial  Parallax. 


Lat. 

Red. 
of  par 

Red.  of 
Lat. 

Eq.Par  Semidia. 

Eq.Par 

Semidia 

Eq.Par 

Semidia. 

sec'  Pro. 
Par. 

,/ 

, 

„ 

/     " 

/      // 

/    // 

, 

n 

'  ,, 

0 

7T~ 

~     ~/7~ 

53     0 

14 

26.5 

56    0 

15  15.6 

59     0 

16 

4.6 

1 

0.3 

0 

0.0 

0    0.0 

53  10 

14 

29.3 

56  10 

15   18.3 

59   10 

16 

7.4 

2 

0.5 

3 

0.0 

1  11.8 

53  20 

14 

32.0 

56  20 

15  21.0 

59  20 

16  10.1 

3 

0.8 

6 

0.1 

2  22.7 

53  30 

14 

34.7 

56  30 

15  23.8 

59  30 

16  12.8 

4 

1.1 

9 

0.3 

3  32.1 

53  40 

14 

37.4 

56  40 

15  26.5 

59  40 

16  15.6 

5 

1.4 

12 
15 

0.5 
0.7 

4  39.3 
5  43.4 

53  50 
54     0 

14 

14 

40.2 
42.9 

56  50 
57     0 

15  29.2 
15  31.9 

59  50 
60     0 

16  18.3 
16  21.0 

6 

7 

1.6 
1.9 

18 

1.0 

6  43.7 

54  10 

14 

45.6 

57  10 

15  34.7 

60  10 

16  23.7 

8 

2.2 

21 

1.4 

7  39.7 

54  20 

14 

48.3 

57  20 

15  37.4 

60  20 

16  26.4 

9 

2.4 

24 

1.8 

8  30.7 

54  30 

14 

51.1 

57  30 

15  40.1 

GO  30 

16  29.2 

10 

2.7 

27 
30 

2.3 

2.7 

9  16.1 
9  55.4 

54  40 
54  50 

14 
14 

53.8 
56.5 

57  40 
57  50 

15  42.8 
15  45.6 

60  40 
60  50 

16  31.9 
16  34.6 

33 

3.3 

10  28.3 

55     0 

14 

59.2 

58     0 

15  48.3 

61     0 

16  37.3 

36 

3.8 

10  54.3 

55  10 

15 

2.0 

58  10 

15  51.0 

61   10 

16  40.1 

39 

4.4 

11  13.2 

55  20 

15 

4.7 

58  20 

15  53.7 

61  20 

16  42.8 

42 
45 

4.9 
5.5 

11  24.7 
11  28.7 

55  30 
55  40 

15 
15 

7.4 
10.1 

58  30 
58  40 

15  56.5 
15  59.2 

61  30 
61  40 

16  45.5 
16  48.2 

48 

6.1 

11  25.2 

55  50 

15 

12.9 

58  50 

16     1.9 

61  50 

16  51.0 

51 

6.7 

11  14.1 

56     0 

15 

15.6 

59     0 

16     4.6 

62     0 

16  53.7 

54 

7.2 

10  55.7 

57 

7.8 

10  30.0 

60 

8.3 

9  57.4 

63 

8.8 

9  18.3 

66 

9.2 

8  32.9 

69 

9.7 

7  42.0 

TABLE  LXVI. 

72 

10.0 

6  45.9 

75 

10.3 

5  45.4 

Augmentation  of  Moon's  Semi-diameter. 

78 
81 

10.6 
10  8 

4  41.0 
3  33  5 

//^ 

84 

1LO 

2  23.7 

A  li. 

Horizon.  Semi-diameter 

Alt 

Horizon.    Semi-diameter. 

87 
90 

11.1 
11.1 

1  12.3 
0     0.0 

Alt. 

14'30" 

15' 

16' 

17 

Alt. 

14'  30" 

15' 

16'      17 

Subsidiary  Table. 

0 

2 

0.6 

0.6 

0.7 

0.8 

42 

9.2 

9.8 

11.2    12.6 

Lat. 

+  3' 

—  3' 

4 

1 

.0 

1.1 

1.3 

1.5 

45 

9.7 

10.4 

11.8    13.3 

6 

1 

.5 

1.6 

1.9 

2.1 

48 

10.2 

10.9 

12.4    14.0 

o 

>/ 

" 

8 

2.0 

2.1 

2.4 

2.7 

51 

10.6 

11.4 

13.0    14.7 

0 

+  0.0 

—  0.0 

10 

2.4 

2.6 

3.0 

3.4 

54 

11.1 

11.8 

13.5    15.2 

$ 

0  0 

0.0 

12 

v.v 

0  0 

0.0 

12 

2.9 

3.1 

3.6 

4.0 

57 

11.5 

12.3 

14.0    15.8 

i& 
15 

U.v 

0.0 

0.0 

14 

3.4 

3.6 

4.1 

4.7 

60 

11  8 

12.7 

14.4    16.3 

A  1 

0.1 

16 

3.8 

4.1 

4.7 

5.3 

63 

12.2 

13.0 

14.9    16.8 

24 

V.  1 

0.1 

0.1 

18 

4.3 

4.6 

5.2 

5.9 

66 

12.5 

13.4 

15.2    17.2 

21 

4.9 

5.3 

6.0 

6.8 

69 

12.8 

13.7 

15.6    17.6 

qn 

0  1 

A  1 

ou 
36 

U.  1 

0.2 

V.  1 

0.2 

24 

5.6 

6.0 

6.8 

7.7 

72 

13.0 

13.9 

15.9    17.9 

42 

A  0 

0.2 

27 

6.2  ' 

6.7 

7.6 

8.6 

75 

13.2 

14.1 

16.1    18.2 

48 

v.«« 

0.3 

0.3 

30 

6.9 

7.3 

8.4 

9.5 

78 

13.4 

14.3 

16.3    18.4 

54 

0.3 

0.3 

33 

7.5 

8.0 

9.1 

10.3 

81 

13.5 

14.4 

16.5    18.6 

36 

8 

1 

8.6 

9.8    11.1 

84 

13.6 

14.5 

16.6    18.7 

60 

Mft 

0.4 

s\  c 

0.4 

39 

8.6 

9.2 

10.5  i  11.9 

90 

13.7 

14.6 

16.7    18.8 

78 

U.5 

0.6 

0.5 
0.6 

84 

0.6 

0.6 

90 

+  0.6    —0.6 

TABLE   LXVII. 


87 


Moon's  Horary  Motion  in  Longitude. 
Arguments.  1  to  18  of  Longitude. 


Arg 

2 

3 

4 

5 

6 

1 

7 

8 

9 

Arg. 

0 

~r~" 

100 

5.0 

0.0 

2.9 

1.9 

0.0 

0.00 

0.00 

0.00 

0.16 

2 

5.0 

0.0 

2.8 

1.9 

0.0 

0.00 

0.00 

0.00 

0.15 

98 

4 

4.9 

0.0 

2.8 

1.9 

0.0 

0.01 

0.00 

0.02 

0.15 

96 

6 

4.8 

0.1 

2.8 

1.9 

0.1 

0.03 

0.01 

0.05 

0.14 

94 

8 

4.7 

0.2 

2.7 

1.8 

0.1 

0.06 

0.01 

0.09 

0.12 

92 

10 

4.5 

0.3 

2.6 

1.7 

0.2 

0.09 

0.02 

0.14 

0.10 

90 

12 

4.3 

0.4 

2.5 

1.7 

0.2 

0.13 

0.02 

0.19 

0.09 

88 

14 

4.1 

0.6 

2.3 

1.6 

0.3 

0.18 

0.03 

0.26 

0.07 

86 

16 

3.8 

0.7 

2.2 

1.5 

0.4 

0.23 

0.04 

0.33 

0.05 

84 

18 

3.6 

0.9 

2.0 

1.4 

0.5 

0.28 

0.05 

0.41 

0.03 

82 

20 

3.3 

1.1 

1.9 

1.3 

0.6 

0.34 

0.06 

0.50 

0.02 

80 

22 

3.0 

1.3 

.7 

1.1 

0.7 

0.40 

0.07 

0.58 

0.01 

78 

24 

2.7 

1.5 

.5 

1.0 

0.8 

0.46 

0.08 

0.67 

0.00 

76 

26 

2.3 

1.7 

.3 

0.9 

0.9 

0.52 

0.10 

0.77 

0.00 

74 

28 

2.0 

1.9 

.2 

0.8 

.0 

0.58 

0.11 

0.86 

0.00 

72 

30 

1.7 

2.1 

.0 

0.7 

.1 

0.63 

0.12 

0.94 

0.01 

70 

32 

1.4 

2.2 

0.8 

0.5 

.2 

0.69 

0.13 

1.03 

0.01 

68 

34 

1.2 

2.4 

0.7 

0.4 

.3 

0.74 

0.14 

1.11 

0.03 

66 

36 

0.9 

2.6 

0.5 

0.3 

.3 

0.78 

0.15 

1.18 

0.05 

64 

38 

0.7 

2.7 

0.4 

0.3 

.4 

0.82 

0.16 

1.25 

0.06 

62 

40 

0.5 

2.8 

0.3 

0.2 

.5 

0.86 

0.16 

1.3Q 

0.08 

60 

42 

0.3 

2.9 

0.2 

0.1 

.5 

0.89 

0.17 

1.35 

0.10 

58 

44 

0.2 

3.0 

0.1 

0.1 

.6 

0.91 

0.17 

1.39 

0.11 

56 

46 

0.1 

3.1 

0.0 

0.0 

.6 

0.93 

0.18 

1.42 

0.12 

54 

48 

0.0 

3.1 

0.0 

0.0 

.6 

0.94 

0.18 

1.44 

0.13 

52 

50 

0.0 

3.1 

0.0 

0.0 

1.6 

0.94 

0.18 

1.44 

0.13 

50 

Arg. 

10 

11 

12 

13 

14 

15 

16 

17 

18 

Alg. 

0 

100 

0.00 

0.26 

0.00 

0.00 

0.00 

0.00 

0.26 

0.00 

0.21 

2 

0.00 

0.25  0.00 

000 

0.00 

0.00 

0.26 

0.00 

0.20 

98 

4 

0.02 

0.24|  0.01 

0.00 

0.01 

0.00 

0.26 

0.00 

0.20 

96 

6 

0.04 

0.22 

0.03 

0.01 

O.C2 

0.01 

0.25 

0.00 

0.20 

94 

8 

0.08 

0.20 

0.04 

0.02 

0.04 

0.01 

0.25 

0.0  1 

0.20 

92 

10 

0.12 

0.17 

0.07 

0.03 

0.06 

0.02 

0.24 

0.01 

0.20 

90 

12 

0.16 

0.14 

0.09 

0.04 

0.09 

0.02 

0.22 

0.02 

0.19 

88 

14 

0.20 

0.11 

0.12 

0.06 

0.12 

0.03 

0.21 

0.02 

0.19 

86 

16 

0.24 

0.08 

0.16 

0.07 

0.15 

0.04 

0.20 

0.03 

0.18 

84 

18 

0.28 

0.05 

0.19 

0.09 

0.19 

0.05 

0.19 

0.04 

0.18 

82 

20 

0.31 

0.03 

0.23 

0.11 

0.22 

0.06 

0.17 

0.05 

0.17 

80 

22 

0.34 

0.01 

0.27 

0.13 

0.26 

0.07 

0.15 

0.06 

0.17 

78 

24 

0.35 

0.00 

0.31 

0.15 

0.30 

0.08 

0.14 

0.07 

0.16 

76. 

26 

0.36 

0.00 

035 

0.17 

0.34 

0.08 

0.12 

0.07 

0.16 

74 

28 

0.35 

0.01 

0.39 

0.19 

0.38 

0.09 

0.11 

0.08 

0.15 

72 

30 

0.34 

0.02 

0.43 

0.21 

0.42 

0.10 

0.09 

0.09 

0.15 

70 

32 

0.32 

0.04 

0.47 

0.23 

0.45 

0.11 

0.07 

0.10 

0.14 

68 

34 

0.29 

0.06 

0.50 

0.25 

0.49 

0.12 

0.06 

0.11 

0.14 

66 

36 

0.26 

0.09 

0.54 

0.26 

0.52 

0.13 

0.05 

0.12 

0.13 

64 

38 

0.22 

0.11 

0.57 

0.28 

0.55 

0.14 

0.04 

0.12 

0.13 

62 

40 

0.18 

0.14 

0.59 

0.29 

0.58 

0.14 

0.02 

0.13 

0.12 

60 

42 

0.15 

0.16 

0.62 

0.30 

0.60 

0.15 

0.01 

0.13 

0.12 

58 

44 

0.12 

0.19 

0.63 

0.31  !  0.62 

0.15 

0.01 

0.14 

0.12 

56 

46 

0.10 

021 

0.65 

0.32  '  0.63 

0.16 

0.00 

0.14  0.12 

54 

48 

0.09 

0.22 

0.6610.32  0.64 

0.16 

0.00 

0.14.  '0.12 

52 

50 

0.08 

0.22  0.66  '  0.32  0.64 

0.16  0.00 

0.14  0.11 

50 

TABLE  LXVIII. 

Moon's  Horary  Motion  in  Longitude. 
Argument.     Argument  of  the  Evection. 


0* 

I* 

II* 

III* 

IV* 

V* 

0 

0 

80.3 

74.7 

59.6 

39.4 

19.8 

5.9 

o 
30 

1 

80.3 

74.3 

58.9 

38.7 

19.3 

5.6 

29 

2 

80.3 

73.9 

58.3 

38.0 

18.7 

5.3 

28 

3 

80.2 

73.5 

57.7 

37.3 

18.1 

5.0 

27 

4 

80.2 

73.1 

57.1 

36.6 

17.6 

4.7 

26 

5 

80.1 

72.7 

56.4 

36.0 

17.0 

4.4 

25  . 

6 

80.1 

72.3 

55.8 

35.3 

16.5 

4.1 

24 

7 

80.0 

71.9 

55.1 

34.6 

15.9 

3.8 

23 

8 

79.9 

71.4 

54.5 

33.9 

15.4 

3.6 

22 

9 

79.8 

71.0 

53.8 

33.2 

14.9 

3.4 

21 

10 

79.7 

70.5 

53.1 

32.5 

14.4 

3.1 

20 

11 

79.5 

70.1 

52.5 

31.9 

13.9 

2.9 

19 

12 

79.4 

69.6 

51.8 

31.2 

13.4 

2.7 

18 

13 

79.2 

69.1 

51.1 

30.5 

12.9 

2.5 

17 

14 

79.1 

68.6 

50.5 

29.9 

12.4 

2.3 

16 

15 

78.9 

68.1 

49.8 

29.2 

11.9 

2.1 

15 

16 

78.7 

67.6 

49.1 

28.6 

11.4 

2.0 

14 

17 

78.5 

67.0 

48.4 

27.9 

11.0 

1.8 

13 

18 

78.2 

66.5 

47.7 

27.2 

10.5 

1.7 

12 

19 

78.0 

66.0 

47.0 

26.6 

10.1 

1.6 

11 

20 

77.8 

65.4 

46.4 

26.0 

9.7 

1.4 

10 

21 

77.5 

64.9 

45.7 

25.3 

9.3 

1.3 

9 

"22 

77.2 

64.3 

45.0 

24.7 

8.8 

1.2 

8 

23 

77.0 

63.7 

44.3 

24.1 

8.4 

1.2 

7 

24 

70.7 

63.2 

43.6 

23.5 

8.0 

1.1 

6 

25 

76.4 

62.6 

42.9 

22.8 

7.7 

1.0 

5 

26 

76.1 

62.0 

42.2 

22.2 

7.3 

1.0 

4 

27 

75.7 

61.4 

41.5 

21.6 

6.9 

0.9 

3 

28 

75.4 

60.8 

40.8 

21.0 

6.6 

0.9 

2 

29 

75.0 

60.2 

40.1 

20.4 

6.2 

0.9 

1 

30 

74.7 

59.6 

39.4 

19.8 

5.9 

0.9 

0 

XI* 

X» 

IX* 

VIII* 

VII* 

VI* 

TABLE  LXIX. 

Moon's  Horary  Motion  in  Longitude. 

Arguments.     Sum  of  Equations,  2,  3,  &c.,  and  Evection  corrected 
{    0"  |    10"  |    20"  | 


*          o 

s     ° 

0        0 

00 

0.2 

0.5 

XII    0 

I         0 

0.0 

0.2 

0.4 

XI      0 

II       0 

0.1 

0.2 

0.3 

X       0 

III      0 

0.2 

0.2 

0.2 

IX      0 

IV      0 

0.3 

0.2 

0.1 

VIII   0 

V       0 

0.4 

0.2 

0.0 

VII     0 

VI      0 

0.5 

0.2 

0.0 

VI      0 

0"    [     10"   I    20"  | 


TABLE   LXX. 


Moon's  Horary  Motion  in  Longitude. 
Arguments.     Sum  of  preceding  equations,  and  Anomaly  corrected. 


°" 

10" 

20" 

30" 

40" 

50" 

60" 

70" 

80" 

90" 

100" 

s   ° 

s  ° 

0  0 

4.1 

5.3 

6.5 

7.6 

8.8 

10.0 

11.2 

12.4 

13.5 

14.7 

15.9 

XII  0 

5 

4.1 

5.3 

6.5 

7.7 

8.8 

10.0 

11.2 

12.3 

13.5 

14.7 

15.9 

25 

10 

4.2 

5.4 

6.5 

7.7 

8.8 

10.0 

11.2 

12.3 

13.5 

14.6 

15.8 

20 

15 

4.3 

5.5 

6.6 

7.7 

8.9 

10.0 

11.1 

12.3 

13.4 

14.5 

15.7 

15 

20 

4.5 

5.6 

6.7 

7.8 

8.9 

10.0 

11.1 

12.2 

13.3 

14.4 

15.5 

10 

25 

4.8 

5.8 

6.9 

7.9 

9.0 

10.0 

11.0 

12.1 

13.1 

14.2 

15.2 

5 

I  -0 

5.1 

6.0 

7.0 

8.0 

9.0 

10.0 

11.0 

12.0 

13.0 

14.0 

14.9 

XI   0 

5 

5.4 

6.3 

7.2 

8.2 

9.1 

10.0 

10.9 

11.8 

12.8 

13.7 

14.6 

25 

10 

5.7 

6.6 

7.4 

8.3 

9.2 

10.0 

10.8 

11.7 

12.6 

13.4 

14.3 

20 

15 

6.1 

6.9 

7.7 

8.5 

9.2 

10.0 

10.8 

11.5 

12.3 

13.1 

13.9 

15 

20 

6.6 

7.2 

7.9 

8.6 

9.3 

10.0 

10.7 

11.4 

12.1 

12.8 

13.4 

10 

25 

7.0 

7.6 

8.2 

8.8 

9.4 

10.0 

10.6 

11.2 

11.8 

12.4 

13.0 

5 

II  0 

7.5 

8.0 

8.5 

9.0 

9.5 

10.0 

10.5 

11.0 

11.5 

12.0 

12.5 

X   0 

5 

,  7.9 

8.4 

8.8 

9.2 

9.6 

10.0 

10.4 

10.8 

11.2 

11.6 

12.1 

25 

10 

8.4 

8.7 

9.1 

9.4 

9.7 

10.0 

10.3 

10.6 

10.9 

11.3 

11.6 

20 

15 

8.9 

9.1 

9.4 

9.6 

9.8 

10.0 

10.2 

10.4 

10.6 

10.9 

11.1 

15 

20 

9.4 

9.5 

9.7 

9.8 

9.9 

10.0 

10.1 

10.2 

10.3 

10.5 

10.6 

10 

25 

9.9 

9.9 

9.9 

10.0 

10.0 

10.0 

10.0 

10.0 

10.1 

10.1 

10.1 

5 

III  0 

10.4 

10.3 

10.2 

10.1 

10.1 

10.0 

9.9 

9.9 

9.8 

9.7 

9.6 

IX   0 

5 

10.8 

10.7 

10.5 

10.3 

10.2 

10.0 

9.8 

9.7 

9.5 

9.3 

9.2 

S5 

10 

11.3 

11.0 

10.8 

10.5 

10.3 

10.0 

9.7 

9.5 

9.2 

9.0 

8.7 

20 

15 

11.7 

11.4 

11.0 

10.7 

10.3 

10.0 

9.7 

9.3 

9.0 

8.6 

8.3 

15 

20 

12.1 

11.7 

11.3 

10.9 

10.4 

10.0 

9.6 

9.1 

8.7 

8.3 

7.9 

10 

25 

12.5 

12.0 

11.5 

11.0 

10.5 

10.0 

9.5 

9.0 

8.5 

8.0 

7.5 

5 

IV  0 

12.9 

12.3 

11.7 

11.2 

10.6 

10.0 

9.4 

8.8 

8.3 

7.7 

7.1 

VIII  0 

5 

13.3 

12.6 

11.9 

11.3 

10.6 

10.0 

9.4 

8".  7 

8.1 

7.4 

6.7 

25 

10 

13.6 

12.9 

12.1 

11.4 

10.7 

10.0 

9.3 

8.6 

7.9 

7.1 

6.4 

20 

15 

13.9 

13.1 

12.3 

11.5 

10.8 

10.0 

9.2 

8.5 

7.7 

6.9 

6.1 

15 

20 

14.1 

13.3 

12.5 

11.6 

10.8 

10.0 

9.2 

8.4 

7.5 

6.7 

5.9 

10 

25 

14.4 

13.5 

12.6 

11.7 

10.9 

10.0 

9.1 

8.3 

7.4 

6.5 

5.6 

5 

V   0 

14.6 

13.7 

12.7 

11.8 

10.9 

10.0 

9.1 

8.2 

7.3 

6.3 

5.4 

VII  0 

5 

14.7 

13.8 

12.8 

11.9 

10.9 

10.0 

9.1 

8.1 

7.2 

6.2 

5.3 

25 

10 

14.9 

13.9 

12.9 

12.0 

11.0 

10.0 

9.0 

8.0 

7.1 

6.1 

5.1 

20 

15 

15.0 

14.0 

13.0 

12.0 

11.0 

10.0 

9.0 

8.0 

7.0 

6.0 

5.0 

15 

20 

15.1 

14.1 

13.0 

12.0 

11.0 

10.0 

9.0 

8.0 

7.0 

5.9 

4.9 

10 

25 

15.1 

14.1 

13.1 

12.0 

11.0 

10.0 

9.0 

8.0 

6.9 

5.9 

4.9 

5 

VI  0 

15.1 

14.1 

13.1 

12.1 

11.0 

10.0 

9.0 

8.0 

6.9 

5.9 

4.9 

VI   0 

0' 

10" 

20" 

30" 

40" 

50" 

60" 

70" 

80" 

90" 

100" 

90  TABLE  LXXI. 

Moon's  Horary  Motion  in  Longitude. 
Argument.     Anomaly  corrected. 


0 

diff. 

I* 

diff.      II*    diff 

III* 

diff. 

IV* 

diff. 

V« 

diff. 

0 

„ 

„ 

// 

ft 

/, 

„ 

0 

0 

1 

2 
3 

4 
5 

441.5 
441.5 
441.3 
441.1 
440.8 
440.4 

0.0 
0.1 
0.2 
0.3 
0.4 

404.1 
401.6 
399.2 
396.6 
394.0 
391.3 

2.5 
2.4 
2.6 
2.6 
2.7 

309.3 
305.6 
301.9 
298.1 
294.4 
290.6 

3.7 
3.7 

3.8  ' 
3.7 
3.8  ! 

195.3 
191.6 

187.9 
184.3 
180.6 
177.0 

3.7 
3.7 
3.6 
3.7 
3.6 

95.8 
93.0 
90.2 
87.6 
84.9 
82.3 

2.8 
2.8 
2.6 
2.7 
2.6 

30.6 
29.2 
27.8 
26.4 
25.1 
23.8 

1.4 
1.4 
1.4 
1.3 
1.3 

30 
29 
28 
27 
26 
25 

0.5 

2.7 

3.8 

3.6 

2.6 

1.2 

6 
7 

8 
9 
10 

439.9 
439.4 
438.7 
438.0 
437.2 

0.5 

0.7 
0.7 
0.8 

388.6 
385.8 
383.0 
380.1 
377.1 

2.8 
2.8 
2.9 
3.0 

286.8 
283.0 
279.2 
275.4 
271.5 

3.8 
3.8 
3.8 
3.9 

173.4 
169.8 
166.3 
162.8 
159.3 

3.6 
3.5 
3.5 
3.5 

79.7 
77.1 
74.6 
72.1 
69.7 

2.6 
2.5 
2.5 
2.4 

22.6 
21.4 
20.3 
19.2 
18.2 

1.2 
1.1 
1.1 
IrO 

24 
23 
22 
21 
20 

0.9 

13.0 

3.8 

3.5 

2.4 

1.0 

11 
12 
13 
14 
15 

436.3 
435.3 
434.2 
433.1 
431.8 

1.0 
1.1 
1.1 
1.3 

374.1 
371.1 
368.0 
364.8 
361.6 

3.0 
3.1 
3.2 
3.2 

267.7 
263.8 
260.0 
256.2 
252.3 

3.9 
3.8 
3.8 
3.9 

155.8 
152.4 
148.9 
145.5 
142.2 

3.4 
3.5 
3.4 
3.3 

67.3 
65.0 
62.7 
60.4 

58.2 

2.3 
2.3 
2,3 

2.2 

17.2 
16.3 
15.4 
14.6 
13.8 

0» 
0.9 
0.8 
0.8 

19 
18 
17 
16 
15 

1.3 

3.2 

3.8 

3.3 

2.1 

0.7 

16 
17 

18 
19 
20 

430.5 
429.1 
427.6 
426.1 
424.5 

1.4 
1.5 
1.5 
1.6 

358.4 
355.1 
351.8 
348.4 
345.0 

3.3 
3.3 
3.4 
3.4 

248.5 
244.6 
240.8 
236.9 
233.1 

3.9 
3.8 
3.9 
3.8 

138.S 
135.6 
132.3 
129.1 
125.9 

3.3 
3.3 
3.2 
3.2 

56.1 
53.9 
51.9 
49.8 
47.9 

2.2 
2.0 
2.1 
1.9 

13.1 
12.4 
11.8 
11.2 
10.7 

0.7 
0.6 
0.6 
0.5 

14 
13 
12 
11 
10 

1.7 

3.4 

3.8 

3.2 

2.0 

0.5 

21 
22 
23 
24 
25 

422.7 
421.0 
419.1 
417.2 
415.2 

1.7 
1.9 
1.9 
2.0 

341.6 
338.1 
334.6 
331.1 
327.5 

3.5 
3.5 
3.5 
3,6 

229.3 
225.4 
221.6 
217.8 
214.0 

3.9 
3.8 
3.8 
3.8 

122.7 
119.6 
116.5 
113.4 
110.4 

3.1 
3.1 
3.1 
3.0 

45.9 
44.0 
42.2 
40.4 
38.7 

1.9 
1.8 
1.8 
1.7 

10.2 
9.8 
9.4- 
9.1 
8.8 

0.4 
0.4 
0.3 
0.3 

9 
8 
7 
6 
5 

2.1 

3.5 

3.7 

3.0 

1.7 

0.2 

26 
27 
28 
29 
30 

413.1 
410.9 
408.7 
406.4 
404.1 

2.2 
2.2 
2.3 
2.3 

324.0 
320.3 
316.7 
313.0 
309.3 

3.7 
3.6 
3.7 
3.7 

210.3 
206.5 
202.8 
199.0 
195.3 

3.8 
3.7 
3.9 
3.7 

107.4 
104.5 
101.6 
98.7 
95.8 

2.9 
2.9 
2.9 
2.9 

37.0 
35.3 
33.7 
32.1 
30.6 

1.7 
1.6 
1.6 
1.5 

8.6 

8.4 
8.3 
8.2 
8.2 

0.2 
0.1 
0.1 
0.0 

4 
3 
2 
1 
0 

XI* 

X* 

IX* 

VIII* 

VII« 

VI* 

TABLE  LXXII. 

Moon's  Horary  Motion  in  Longitude. 
Arguments.   Sum  of  preceding  Equations,  and  Arg.  of  Variation. 


91 


0 

50 

100 

150 

200 

250 

300 

350 

400 

450 

500 

550 

600 

a  o 
0  0 

4.5 

5.5 

6.5 

7.6 

8.6 

9.6 

10.6 

11.6 

12.6 

13.7 

14.7 

15.7 

16.7 

8   ° 

XII  0 

5 

4.6 

5.6 

6.6 

7.6 

8.6 

9.6 

10.6 

11.6 

12.6 

13.6 

14.6 

15.6 

16.6 

25 

10 

4.8 

5.8 

6.8 

7.7 

8.7 

9.6 

10.6 

11.5 

12.5 

13.4 

14.4 

15.3 

16.3 

20 

15 

5.3 

6.1 

7.0 

7.9 

8.8 

9.7 

10.5 

11.4 

12.3 

13.1 

14.0 

14.9 

15.8 

15 

20 

5.8 

6.6 

7.4 

8.2 

8.9 

9.7 

10.5 

11.2 

12.0 

12.8 

13.5 

14.3 

15.1 

10 

25 

6.6 

7.2 

7.8 

8.5 

9.1 

9.7 

10.4 

11.0 

11.7 

12.3 

12.9 

13.6 

14.2 

5 

I   0 

7.4 

7.8 

8.3 

8.8 

9.3 

9.8 

10.3 

10.8 

11.3 

11.8 

12.3 

12.7 

13.2 

XI  0 

5 

8.3 

8.6 

8.9 

9.2 

9.5 

9.9 

10.2 

10.5 

10.8 

11.2 

11.5 

11.8 

12.1 

25 

10 

9.2 

9.3 

9.5 

9.6 

9.8 

9.9 

10.1 

10.2 

10.4 

10.5 

10.7 

10.8 

11.0 

20 

15 

10.2 

10.1 

10.1 

10.1 

10.0 

10.0 

10.0 

10.0 

9.9 

9.9 

9.9 

9.8 

9.8 

15 

20 

11.1 

10.9 

10.7 

10.5 

10.3 

10.1 

9.9 

9.7 

9.5 

9-2 

9.0 

8.8 

8.6 

10 

25 

12.1 

11.7 

11.3 

10.9 

10.5 

10.2 

9.8 

9.4 

9.0 

8.6 

8.3 

7.9 

7.5 

5 

II  0 

12.9 

12.4 

11.8 

11.3 

10.8 

10.2 

9.7 

9.1 

8.6 

8.1 

7.5 

7.0 

6.4 

X  0 

5 

13.7 

13.0 

12.3 

11.6 

11.0 

10.3 

9.6 

8.9 

8.2 

7.5 

6.9 

6.2 

5.5 

25 

10 

14.3 

13.5 

12.7 

11.9 

11.1 

10.3 

9.5 

8.7 

7.9 

7.1 

6.3 

5.5 

4.7 

20 

15 

14.9 

14.0 

13.1 

12.2 

11.3 

10.4 

9.5 

8.6 

7.7 

6.8 

5.8 

4.9 

4.0 

15 

20 

15.3 

14.3 

13.3 

12.3 

11.4 

10.4 

9.4 

8.4 

7.5 

6.5 

5.5 

4.5 

3.6 

10 

25 

15.5 

14.5 

13.5 

12.4 

11.4 

10.4 

9.4 

8.4 

7.4 

6.3 

5.3 

4.3 

3.3 

5 

III  0 

15.6 

14.5 

13.5 

12.5 

11.4 

10.4 

9.4 

8.4 

7.3 

6.3 

5.3 

4.2 

3.2 

IX  0 

5 

15.4 

14.4 

13.4 

12.4 

11.4 

10.4 

9.4 

8.4 

7.4 

6.4 

5.4 

4.4 

3.3 

25 

10 

15.2 

14.2 

13.3 

12.3 

11.3 

10.4 

9.4 

8.5 

7.5 

6.5 

5.6 

4.6 

3.6 

20 

15 

14.8 

13.9 

13.0 

12.1 

11.2 

10.4 

9.5 

8.6 

7.7 

6.8 

5.9 

5.1 

4.2 

15 

20 

14.2 

13.4 

12.6 

11.9 

11.1 

10.3 

9.5 

8.8 

8.0 

7.2 

6.4 

5.6 

4.9 

10 

25 

13.5 

12.9 

12.2 

11.6 

10.9 

10.3 

9.6 

9.0 

8.4 

7.6 

7.0 

6.3 

5.7 

5 

IV  0 

12.7 

12.2 

11.7 

11.2 

10.7 

10.2 

9.7 

9.2 

8.7 

8.2 

7.7 

7.2 

6.7 

YIIIO 

5 

11.9 

11.5 

11.2 

10.8 

10.5 

10.1 

9.8 

9.5 

9.1 

8.8 

8.4 

8.1 

7.7 

25 

10il0.9 

10.7 

10.6 

10.4 

10.2 

10.1 

9.9 

9.7 

9.6 

9.4 

9.2 

9.1 

8.9 

20 

15 

9.9 

99 

10.0 

10.0 

10.0 

10.0 

10.0 

10.0 

10.0 

10.0 

10.1 

10.1  10.1 

15 

20 

8.9 

9.1 

9.3 

9.5 

9.7 

9.9 

10.1 

10.3 

10.5 

10.7 

10.9 

11.1 

11.3 

10 

25 

8.0 

8.4 

8.7 

9.1 

9.5 

9.9 

10.2 

10.6 

11.0 

11.3 

11.7 

12.1 

12.5 

5 

V  0 

7.1 

7.6 

8.2 

8.7 

9.2 

9.8 

10.3 

10.9 

11.4 

11.9 

12.5 

13.0 

13.6 

VII  0 

5 

6.3 

7.0 

7.6 

8.3 

9.0 

9.7 

10.4 

11.1 

11.8 

12.5 

13.2 

13.9 

14.6 

25 

10 

5.6 

6.4 

.7.2 

8.0 

8.8 

9.7 

10.5 

11.3 

12.1 

13.0 

13.8 

14.6 

15.4 

20| 

15 

5.0 

5.9 

6.8 

7.8 

8.7 

9.6 

10.6 

11.5 

12.4 

13.3 

14.3 

15.2 

16.1 

15 

20 

4.6 

5.6 

6.6 

7.6 

8.6 

9.6 

10.6 

11.6 

12.6 

13.6 

14.6 

15.7 

16.7 

10 

25 

4.3 

5.4 

6.4 

7.5 

8.5 

9.6 

10.6  11.7 

12.7 

13.8 

14.9 

15.9 

17.0 

5 

VI  0 

4.2 

5.3 

6.4 

7.4 

8.5 

9.6 

10.6  11.7 

12.8 

13.9 

14.9 

16.0 

17.1 

VI  0 

0 

50 

100 

150 

200 

250 

300  350 

400 

450 

500 

550 

600 

TABLE  LXXIII. 

Moon's  Horary  Motion  in  Longitude. 
Argument.     Argument  of  the  Variation. 


O 

I* 

II* 

III* 

IV* 

V* 

o 

„ 

// 

// 

// 

// 

// 

0 

0 

77.2 

57.8 

20.3 

2.4 

21.5 

59.7 

30 

1 

77.2 

56.7 

19.2 

2.5 

22.7 

60.9 

29 

2 

77.1 

55.5 

18.1 

2.6 

23.8 

62.0 

28 

3 

77.0 

54.3 

17.0 

2.7 

25.0 

63.1 

27 

4 

76.8 

53.1 

16.0 

2.9 

26.2 

64.2 

26 

5 

76.6 

51.8 

15.0 

3.1 

27.5 

65.3 

25 

6 

76.4 

50.5 

14.1 

3.3 

28.7 

66.3 

24 

7 

76.1 

49.3 

13.2 

3.7 

30.0 

67.3 

23 

8 

75.7 

48.0 

12.3 

4.0 

31.3 

68.3 

22 

9 

75.3 

46.7 

11.4 

4.4 

32.6 

69.2 

21 

10 

74.9 

45.4 

10.6 

4.9 

33.9 

70.1 

20 

11 

74.4 

44.1 

9.8 

5.3 

35.2 

70.9 

19 

12 

73.9 

42.8 

9.0 

5.9 

36.5 

71.7 

18 

13 

73.3 

41.5 

8.3 

6.4 

37.8 

72.5 

17 

14 

72.7 

40.2 

7.6 

7.0 

39.2 

73.3 

16 

15 

72.0 

38.9 

7.0 

7.7 

40.5 

74.0 

15 

16 

71.3 

37.5 

6.4 

8.3 

41.8 

74.7 

14 

17 

70.6 

38.2 

5.8 

9.1 

43.2 

75.3 

13 

18 

69.8 

34.9 

5.3 

9.8 

44.5 

75.8 

12 

19 

69.0 

33.6 

4.8 

10.6 

45.8 

76.4 

11 

20 

68.1 

32.3 

4.4 

11.5 

47.2 

76.9 

10 

21 

67.2 

31.1 

4.0 

12.3 

48.5 

77.3 

9 

22 

66.3 

29.8 

3.7 

13.2 

49.8 

77.7 

8 

23 

65.3 

28.6 

3.3 

14.2 

51.1 

78.1 

7 

24 

64.4 

27.3 

3.1 

15.1 

52.4 

78.4 

6 

25 

63.4 

26.1 

2.9 

16.1 

53.6 

78.6 

5 

26 

62.3 

24.9 

2.7 

17.1 

54.9 

78.9 

4 

27 

61.2 

23.7 

2.5 

18.2 

56.1 

79.0 

3 

28 

60.1 

22.5 

2.5 

19.3 

57.3 

79.2 

2 

29 

59.0 

21.4 

2.4 

20.4 

58.5 

79.2 

1 

30 

57.8 

20.3 

2.4 

21.5 

59.7 

79.2 

0 

. 

XI* 

X* 

IX« 

VIII* 

VII' 

VI* 

TABLE  LXXIV.  03 

Moon's  Horary  Motion  in  Longitude. 
Arguments.    Arg.  of  Reduction  and  Sum  of  preceding  Equations 


0 

50 

100 

150 

200 

250 

300 

350  400 

450 

500  550 

600 

650 

»  o 
O  0 

3.3 

3.1 

2.9 

2.7 

2.5 

2.3 

2. 

1.9 

1.7 

1.5 

1.3 

1.1 

0.9 

0.7 

XII  0 

5  13.3 

3.1 

2.9 

2.7 

2.5 

2.3 

2. 

1.9 

1.7 

1.5 

1.3 

1.1 

0.9 

0.7 

25 

10  13.2 

3.0 

2.8 

2.6 

2.4 

2.3 

2. 

1.9 

1.7 

1.5 

1.3 

1.1 

1.0 

0.8 

20 

15  |3.1 

2.9 

2.8 

2.6 

2.4 

2.2 

2. 

1.9 

1.7 

1.5 

1.4 

1.2 

1.0 

0.9 

15 

20 

3.0 

2.8 

2.7 

2.5 

2.4 

2.2 

2. 

1.9 

1.8 

1.6 

1.5 

1.3 

1.1 

1.0 

10 

25 

2.8 

2.7 

2.6 

2.4 

2.3 

2.2 

2: 

1.9 

1.8 

1.7 

1.5 

1.4 

1.3 

1.2 

5 

I   0 

2.6 

2.5 

2.4 

2.3 

2.2 

2.1 

2.0 

1.9 

1.8 

1.7 

1.6 

1.5 

1.4 

1.3 

XI  0 

5 

2.4 

2.4 

2.3 

2.2 

2.2 

2.1 

2.0 

2.0 

1.9 

1.8 

1.8 

1.7 

1.6 

1.6 

25 

10 

2.2 

2.2 

2.2 

2.1 

2.1 

2.0 

2.0 

2.0 

1.9 

1.9 

1.9 

1.8 

1.8 

1.8 

20 

15 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

15 

20 

1.8 

1.8 

1.8 

1.9 

1.9 

1.9 

2.0 

2.0 

2.1 

2.1 

2.1 

2.2 

2.2 

2.2 

10 

25 

1.6 

1.6 

1.7 

1.8 

1.8 

1.9 

2.0 

2.0 

2.1 

2.2 

2.2 

2.3 

2.4 

2.4 

5 

II  0 

1.4 

1.5 

1.6 

1.7 

1.8 

1.9 

2.0 

2.1 

2.2 

2.3 

2.4 

2.5 

2.6 

2.7 

X   0 

5 

1.2 

1.3 

1.4 

1.6 

1.7 

1.8 

.9 

2. 

2.2 

2.3 

2.5 

2.6 

2.7 

2.8 

25 

10 

1.0 

1.2 

1.3 

1.5 

1.6 

1.8 

.9 

2. 

2.2 

2.4 

2.5 

2.7 

2.9 

3.0 

20 

15 

0.9 

1.1 

1.2 

1.4 

1.6 

.8 

.9 

2. 

2.3 

2.5 

2.6 

2.8 

3.0 

3.1 

15 

20 

0.8 

1.0 

1.2 

1.4 

1.6 

.7 

.9 

2. 

2.3 

2.5 

2.7 

2.9 

3.0 

3.2 

10 

25 

0.7 

0.9 

1.1 

1.3 

1.5 

.7 

.9 

2. 

2.3 

2.5 

2.7 

2.9 

3.1 

3.3 

5 

III  0 

0.7 

0.9 

1.1 

1.3 

1.5 

.7 

.9 

2. 

2.3 

2.5 

2.7 

2.9 

3.1 

3.3 

IX   0 

5 

0.7 

0.9 

1.1 

1.3 

1.5 

.7 

.9 

2 

2.3 

2.5 

2.7 

2.9 

3.1 

3.3 

25 

10 

0.8 

.0 

1.2 

1.4 

1.6 

.7 

.9 

2 

2.3 

2.5 

2.7 

2.9 

3.0 

3.2 

20 

15 

0.9 

.1 

1.2 

1.4 

1.6 

.8 

.9 

2.1 

2.3 

2.5 

2.6 

2.8 

3.0 

3.1 

15 

20 

1.0 

.2 

1.3 

1.5 

1.6 

.8 

.9 

2.1 

2.2 

2.4 

2.5 

2.7. 

2.9 

3.0 

10 

25 

1.2 

.3 

1.4 

1.6 

1.7 

.8 

.9 

2.1 

2.2 

2.3 

2.5 

2.6 

2.7  2.8 

5 

IV  0 

1.4 

.5 

1.6 

1.7 

1.8 

.9 

2.0 

2.1 

2.2 

2.3 

2.4 

2.5 

2.6^.7 

VIII  0 

| 

.   5 

1.6 

1.6 

1.7 

1.8 

1.8 

.9 

2.0 

2.0 

2.1 

2  2 

2  ° 

2.3 

2.4  2.4 

25 

10 

1.8 

1.8 

1.8 

1.9 

1.9 

.9 

2.0 

2.0 

2.1 

2.1 

2.1 

2.2 

2.2  2.2 

20 

15 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0  2.0 

15 

20 

2.2 

2.2 

2.2 

2.1 

2.1 

2.0 

2.0 

2.0 

1.9 

1.9 

1.9 

.8 

1.8  1.8 

10 

25 

2.4 

2.4 

2.3 

2.2 

2.2 

2.1 

2.0 

2.0 

1.9 

1.8 

1.8 

.7 

1.6 

1.6 

5 

V  0 

2.6 

2.5 

2.4 

2.3 

2.2 

2.1 

2.0 

1.9 

1.8 

1.7 

1.6 

.5 

1.4 

1.3 

Vll  0 

5 

2.8 

2.7 

2.6 

2.4 

2.3 

2.2 

2.1 

1.9 

1.8 

1.7 

1.5 

.4 

1.3 

1.2 

25 

10 

3.0 

2.8 

2.7 

2.5 

2.4 

2.2 

2.1 

1.9 

1.8 

1.6 

1.5 

.3 

1.1 

1.0 

20 

15 

3.1 

2  9 

?  8 

?  6 

*>  4 

0  t> 

?!  1 

1  9 

1  7 

1  *\ 

A 

o 

1  0 

0  9 

15 

20 

3.2 

3.0 

2.8 

2.6 

2.4 

2.3 

2.1 

1.9 

1.7 

1.5 

1.3 

.1 

1.0 

0.8 

10 

25 

3.3 

3.1 

2.9 

2.7 

2.5 

2.3 

2.1 

1.9 

1.7 

1.5 

1.3 

.1 

0.9 

0.7 

5 

VI  0 

3.3 

3.1 

2.9 

2.7 

2.5 

2.3 

2.1 

1.9 

1.7 

1.5 

1.3 

.1 

0.9 

0.7 

VI  0 

0 

50 

100 

150 

200 

250 

300 

350 

400 

450 

500 

550 

600 

650 

94         TABLE  LXXV. 
Moon's  Horary  Motion  in  Long. 
Arg.    Arg.  of  Reduction. 


TABLE  LXXVI. 
Moon's  Horary  Motion  in  Long. 

(Equation  of  the  second  order.) 
Arguments.    Arg's.of  Table  LXX. 


Os  Vis 

I*  VI!s  Us  \Ills 

o 

„ 

II 

// 

0 

0 

2.1 

6.0 

14.0 

30 

1 

2.1 

6.3 

14.2 

29 

2 

2.1 

6.5 

14.4 

28 

3 

2.1 

6.8 

14.7 

27 

4 

2.2 

7.0 

14.9 

26 

5 

2.2 

7.3 

15.1 

25 

6 

2.2 

7.5 

15.3 

24 

7 

2.3 

7.8 

15.5 

23 

8 

2.4 

8.1 

15.7 

22 

9 

2.5 

8.4 

15.9 

21 

10 

2.5 

8.6 

16.1 

20 

11 

2.6 

8.9 

16.2 

19 

12 

2.7 

9.2 

16.4 

18 

13 

2.9 

9.4 

16.6 

17 

14 

3.0 

9.7 

16.7 

16 

15 

3.1 

10.0 

16.9 

15 

16 

3.3 

10.3 

17.0 

14 

17 

3.4 

10.6 

17.1 

13 

18 

3.6 

10.8 

17.3 

12 

19 

3.8 

11.1 

17.4 

11 

20 

3.9 

11.4 

17.5 

10 

21 

4.1 

11.6 

17.5 

9 

22 

4.3 

11.9 

17.6 

8 

23 

4.5 

12.2 

17.7 

7 

24 

4.7 

12.5 

17.8 

6 

25 

4.9 

12.7 

17.8 

5 

26 

5.1 

13.0 

17.8 

4 

27 

5.3 

13.2 

17.9 

3 

28 

5.6 

13.5 

17.9 

2 

29 

5.8 

13.7 

17.9 

1 

|  30 

6.0 

14.0 

17.9 

0 

1 

XT«V» 

X»IV« 

IXsIIIs 

ft 

„ 

„ 

Arg. 

0 

50 

100 

a         ° 

// 

// 

" 

0        0 

0.05 

0.05 

0.05 

I    ,      0 

0.08 

0.05 

0.02 

II        0 

0.10 

0.05 

0.00 

III      0 

0.10 

0.05 

0.00 

IV      0 

0.09 

0.05 

0.01 

V        0 

0.07 

0.05 

0.03 

VI      0 

0.05 

0.05 

0.05 

VII     0 

0.03 

0.05 

0.07 

VIII  0 

0.01 

0.05 

0.09 

IX      0 

0.00 

0.05 

0.10 

X       0 

0.00 

0.05 

0.10 

XI      0 

0.02 

,  0.05 

0.08 

XII    0 

0.05 

0.05 

0.05 

tt 
0 

50 

100 

Constant  to  be  added  27'24".0. 

TABLE  LXXVII. 
Moon's  Horary  Motion  in  Longitude. 

(Equations  of  the  second  order.) 
Arguments.     Arguments  of  Tables  LXXII  and  LXXIV. 


Variation. 

Reduction. 

tt 
0 

100 

200 

300 

400 

500 

600 

0 

tf 
600 

•      •    ° 

0.     VI.     0 

0.14 

0.14 

0.14 

0.14 

0.14 

0.14 

0.14 

003 

0.03 

I.      VII.    0 

0.22 

0.19 

0.16 

0.13 

0.10 

0.06 

0.02 

0.01 

0.05 

I.      VII.  15 

0.23 

0.20 

0.17 

0.13 

0.10 

0.05 

0.01 

0.01 

0.06 

n.   vin.  o 

0.22 

0.19 

0.16 

0.13 

0.10 

0.07 

0.03 

0.01 

0.05 

m.  ix.    o 

0.14 

0.14 

0.14 

0.14 

0.14 

0.14 

0.14 

0.03 

0.03 

IV.    X.       0 

0.06 

0.09 

0.12 

0.15 

0.18 

0.21 

0.26 

0.05 

0.01 

IV.    X.     15 

0.05 

0.08 

0.11 

0.15 

0.18 

0.23 

0.28 

0.05 

0.00 

V.     XI.      0 

0.06 

0.09 

0.12 

0.15 

0.18 

0.22 

0.26 

0.05 

0.01 

VI.    XII.    0 

0.14 

0.14 

0.14  J0.14 

0.14 

C.14 

0.14 

0.03 

0.03 

TABLE  LXXVIII.  95 

Moon's  Horary  Motion  in  Longitude. 

(Equations  of  the  second  order.) 
Arguments.     Args.  of  Evection,  Anomaly,  Variation,  Reduction. 


Evec. 

Anom. 

Var. 

Red. 

Evec. 

Anom. 

Var. 

Red. 

0*       0 

0.16 

1.05 

0.34 

0.08 

0.16 

1.05 

0.34 

0.08 

8         ° 

XII       0  ' 

5 

0.15 

0.93 

0.28 

0.09 

0.18 

1.17 

0.40 

0.06 

25 

10 

0.13 

0.81 

0.22 

0.10 

0.19 

1.28 

0.46 

0.05 

20 

15 

0.12 

0.70 

0.17 

0.11 

0.21 

1.40 

0.51 

0.04 

15 

20 

0.10 

0.59 

0.12 

0.12 

0.22 

1.50 

0.56 

0.03 

10 

25 

0.09 

0.49 

0.08 

0.13 

0.24 

1.60 

0.60 

0.02 

5 

I           0 

0.08 

0.40 

0.05 

0.14 

0.25 

1.70 

0.63 

0.01 

XI        0 

5 

0.07 

0.31 

0.02 

0.15 

0.26 

1.78 

0.66 

0.01 

25 

10 

0.05 

0.24 

0.01 

0.15 

0.27 

1.86 

0.67 

0.00 

20 

15 

0.04 

0.17 

0.01 

0.15 

0.28 

1.92 

0.67 

0.00 

15 

20 

0.03 

0.12 

0.01 

0.15 

0.29 

1.98 

0.67 

0.00 

'*••      10 

25 

0.03 

0.07 

0.03 

0.15 

0.30 

2.02 

0.65 

0.01 

5 

II          0 

0.02 

0.04 

0.06 

0.14 

0.31 

2.05 

0.62 

0.01 

X         0 

5 

0.01 

0.02 

0.09 

0.13 

0.32 

2.08 

0.59 

0.02 

25 

10 

0.01 

0.00 

0.13 

0.12 

0.32 

2.09 

0.54 

0.03 

20 

15 

0.00 

0.00 

0.18 

0.11 

0.32 

2.10 

0.50 

0.04 

15 

20 

0.00 

0.00 

0.24 

0.10 

0.33 

2.09 

0.44 

0.05 

10 

25 

0.00 

0.02 

0.29 

0.09 

0.33 

2.08 

0.39 

0.06 

5 

III        0 

0.00 

0.04 

0.35 

0.08 

0.33 

2.06 

0.33 

0.08 

IX         0 

5 

0.00 

0.07 

0.40 

0.06 

0.33 

2.03 

0.27 

0.09 

25 

10 

0.01 

0.10 

0.46 

0.05 

0.32 

2.00 

0.22 

0.10 

20 

15 

0.01 

0.14 

0.51 

0.04 

0.32 

1.96 

0.17 

.0.11 

15 

20 

0-01 

0.18 

0.56 

0.03 

0.31 

1.91 

0.12  { 

F0.12 

10 

25 

0.02 

0.23 

0.60 

0.02 

0.31 

1.87 

0.08 

0.13 

5 

IV         0 

0.03 

0.28 

0.63 

0.01 

0.30 

1.82 

0.05 

0.14 

VIII     0 

5 

0.03 

0.34 

0.66 

0.01 

0.29 

1.76 

0.02 

0.15 

25 

10 

0.04 

0.39 

0.67 

0.00 

0.28 

1.70 

0.01 

0.15 

20 

15 

0.05 

0.45 

0.68 

0.00 

0.27 

1.64 

0.00 

0.15 

15 

20 

0.06 

0.52 

0.67 

0.00 

0.26 

1.58 

0.00 

0.15 

10 

25 

0.08 

0.58 

0.66 

0.01 

0.25 

1.52 

0.02 

0.15 

5 

V          0 

0.09 

0.64 

0.64 

0.01 

0.24 

1.45 

0.04 

0.14 

VII       0 

5 

0.10 

0.71 

0.60 

0.02 

0.23 

1.39 

0.08 

0.13 

25 

10 

0.11 

0.78 

0.56 

0.03 

0.22 

1.32 

0.12 

0.12 

20 

15 

0.12 

0.84 

0.51 

0.04 

0.20 

1.25 

0.16 

0.11 

lo 

20 

0.14 

0.91 

0.46 

0.05 

0.19 

1.18 

0.22 

0.10 

10 

25 

0.15 

0.98 

0.40 

0.06 

0.18 

1.12 

0.28 

0.09 

5 

VI        0 

0.16 

1.05 

0.34 

0.08 

0.16 

1.05 

0.34 

0.08 

VI         0 

16 


TABLE  LXXIX. 

Moon's  Horary  Motion  in  Latitude. 
Argument.     Arg.  I  of  Latitude. 


0* 

I* 

II* 

III* 

IV* 

V« 

• 

0 

// 

// 

„ 

n 

// 

// 

o 

0 

378.0 

354.3 

289.2 

200.0 

110.8 

45.7 

30 

1 

378.0 

352.7 

286.5 

196.9 

108.1 

44.2 

29 

2 

377.9 

351.1 

283.8 

193.8 

105.4 

42.7 

28 

3 

377.8 

349.4 

281.0 

190.7 

102.8 

41.3 

27 

4 

377.6 

347.7 

278.3 

187.5 

100.2 

39.9 

26 

5 

377.3 

346.0 

275.5 

184.4 

97.7 

38.6 

25 

6 

377.0 

344.2 

272.6 

181.3 

95.1 

37.3 

24 

7 

376.7 

342.3 

269.8 

178.2 

92.6 

36.1 

23 

8        376.3 

340.5 

266.9 

175.1 

90.2 

34.9 

22 

9 

375.8 

338.5 

264.0 

172.1 

87.7 

33.8 

21 

10 

375.3 

336.6 

261.1 

169.0 

85.3 

32.7 

20 

11 

374.7 

334.5 

258.1 

165.9 

83.0 

31.6 

19 

12 

374.1 

332.5 

255.2 

162.9 

80.7 

30.7 

18 

13 

373.5 

330.4 

252.2 

159.8 

78.1 

29.7 

17 

14 

372.7 

328.3 

249.2 

156.8 

76.1 

28.9 

16 

15 

372.0 

326.1 

246.2 

153.8 

73.9 

28.0 

15 

16 

371.1 

323.9 

243.2 

150.8 

71.7 

27.3 

14 

17 

370.3 

321.9 

240.2 

147.8 

69.6 

26.5 

13 

18 

369.3 

319.3 

237.1 

144.8 

67.5 

25.9 

12 

19 

368.4 

317.0 

234.1 

141.9 

65.5 

25.3 

11 

20 

367.3 

314.7 

231.0 

138.9 

63.4 

24.7 

10 

21 

366.2 

312.3 

227.9 

136.0 

61.5  - 

24.2 

9 

22 

365.1 

309.8 

224.9 

133.1 

59.5 

23.7 

8 

23 

363.9 

307.4 

221.8 

130.2 

57.7 

23.3 

7 

24 

362.7 

304.9 

218.7 

127.4 

55.8 

23.0 

6 

25 

361.4 

302.3 

215.6 

124.5 

54.0 

22.7 

5 

26 

360.1% 

299.8 

212.5 

121.7 

52.3 

22.4 

4 

27 

358.7 

297.2 

209.3 

119.0 

50.6 

22.2 

3 

28 

357.3 

294.6 

206.2 

116.2 

48.9 

22.1 

2 

29 

355.8 

291.9 

203.1 

113.5 

47.3 

22.0 

•    1 

30 

354.3 

289.2 

200.0 

110.8 

45.7 

22.0 

0 

XI* 

X* 

IX* 

VIII* 

VII* 

Vis 

TABLE  LXXX. 

Moon's  Horary  Motion  in  Latitude. 
Arguments.  Args.  V,  VI,  VII,  VIII,  IX,  X,  XI,  and  XII,  of  Latitude 


Arg. 

V 

VI 

VII 

VIII 

IX 

X 

XI 

XII 

Arg. 

0 

0.00 

0.50 

0.34 

0.00 

0.50 

0.04 

0.12 

0.08 

1000 

50 

0.01 

0.49 

0.33 

0.00 

0.49 

0.04 

0.12 

0.07 

950 

100 

0.04  0.45 

0.30 

0.02 

0.45 

0.04 

0.11 

0.05 

900 

150 

0.09  0.40 

0.27 

0.04 

0.40 

0.03 

0.10 

0.03 

850 

200 

0.16  0.33 

0.22 

0.06 

0.33 

0.03 

0.08 

0.01 

800 

250 

0.23  0.25 

0.17 

0.09 

0.25 

0.02 

0.06 

0.00 

750 

300 

0.30  0.17 

0.12 

0.12 

0.17 

0.01 

0.04J0.01 

700 

350 

0.37  0.10 

0.07 

0.14 

0.10 

0.01 

0.02  0.03 

650 

400 

0.42  0.05 

0.04 

6.16 

0.05 

0.00 

0.01;0.05 

600 

450 

0.45  0.01 

0.01 

0.18 

0.01 

0,00 

0.00 

0.07 

550 

500 

0.46  0.00'  0.00 

0.18 

0.00 

0.00 

0.00 

0.08 

500 

TABLE  LXXXI.     Moon's  Horary  Motion  in  Latitude.         97 

Arguments.     Preceding  equation,   and  Sum  of  equations   of  Horary 

Motion  in  Longitude,  except  the  last  two. 


Pr. 

eq. 

0" 
1".6 

50" 
1".4 

100" 

150" 
0".9 

200" 

250" 
0".4 

300" 
0".l 

350" 
0''.2 

400" 
0".4 

A50" 

500' 
0".9 

550" 

600" 

^ 

650" 

Dill 

l."l 

0".6 

0".7 

1".2 

1".7 
~ 

' 

20 

59.0 

54.5 

50.0 

45.4 

40.9 

36.4 

31.8 

27.3 

22.8 

18.2 

13.7 

9.1 

4.6 

0.1 

4.5 

30 

57.4 

53.1 

48.9 

44.6 

40.3 

36.0 

31.7 

27.4 

23.2 

18.9 

146 

10.3 

6.0 

1.7 

4.3 

40 

55.8151.8 

47.7 

43.7 

39.7 

35.6 

31.6 

27.6 

23.6 

19.5 

15.5 

11.5 

7.4 

3.1 

4.0 

50 

54.2 

50.4!  46.6 

42.9 

39.1 

35.3 

31.5 

27.7 

24.0 

20.2 

16.4 

12.6 

8.8 

5.1 

3.8 

60 

52.  6  '49.1 

45.5 

42.0 

38.5 

34.9 

31.4 

27.9 

24.4 

20.8 

17.3 

13.8 

10.2 

6.7 

3.5 

70 

51.0 

47.7 

44.4 

41.1 

37.9 

34.6 

31.3 

28.0 

21,8 

21.5 

18.2 

14.9 

11.7 

8.4 

3.3 

80 

49.3 

46.3 

43.3 

40.3 

37.3 

34.2 

31.2 

28.2 

25.2 

22.1 

19.1 

16.1 

13.1   10.0  3.0 

90 

47.7i45.OJ  42.2 

39.4 

36.7 

33.9 

31.1 

28.3 

25.6 

22.8 

20.0 

17.3 

14.5  11.7 

2.8 

100 

46.1 

43.6 

41.1 

38.6 

36.0 

33.5 

31.0 

28.5 

26.0 

23.4 

20.9 

18.4 

15.9   13.4 

2.5 

110 

44.5 

42.2 

40.0 

37.7 

35.4 

33.2 

30.9  28.6 

26.4 

24.1 

21.8 

19.6 

17.3  15.0 

2.3 

120 

42  9 

40.9 

38.9 

36.9 

34.8 

32.8 

30.8 

28.8 

26.8 

24.8 

22.7 

20.7 

18.7  16.7 

2.0 

130 

41.3 

39.5 

37.8 

36.0 

34.2 

32.5 

30.7 

28.9 

27.2 

25.4 

23.7 

21.9 

20.1 

18.4  1.8 

140 

39.7 

38.2 

36.7 

35.1 

33.6 

32.1 

30.6 

29.1 

27.6 

26.1 

24.6 

23.0 

21.5  20.0 

1.5 

150 

38.1 

36.8 

35.5 

34.3 

33.0 

31.8 

30.5 

29.2 

28.0 

26.7 

25.5 

24.2 

23.0  21.7 

1.3 

160 

36.5 

35.4 

34.4 

33.4 

32.4 

31.4 

30.4 

29.4 

28.4 

27.4 

26.4 

25.4 

24.4  23.3 

1.0 

170 

34.8 

34.1 

33.3 

32.6 

31.8 

31.1 

30.3 

29.5 

28.8 

28.0 

27.3 

26.5 

25.8  25.0 

0.8 

180 

33.2 

32.7 

32.2 

31.7 

31.2 

30.7 

30.2 

29.7 

29.2 

28.7 

28.2 

27.7 

27.2  26.7 

0.5 

190 

31.6 

31.4 

31.1 

30.9 

30.6 

30.4 

30.1 

29.8 

29.6 

29.3 

29.1 

28.8 

28.  6|  28.3 

0.3 

200 

30.0 

30.0 

30.0 

30.01  30.0 

30.0 

30.0 

30.0 

30.0 

80.0 

30.0 

30.0 

30.0  30.0 

0.0 

210 

28.4(28.6 

28.9 

29.1 

29.4 

29.6 

29.9 

30.2 

30.4 

30.7 

30.9 

31.2 

31.4  31.7 

0.3 

220 

26.3 

27.3  27.8 

28.3  28.8 

29.3 

29.8  30.3 

30.8 

31.3 

31.8 

32.3 

32.8  33.3 

0.5 

230 

25.2 

25.9,  26.7 

27.4   28.2 

28.9 

29.7 

30.5 

31.2 

32.0 

32.7 

33.5 

34.2 

35.0 

0.8 

240 

23.5  24.6!  25.6 

26.6)  27.6 

28.6 

29.6 

30.6 

31.6 

32.6 

33.6 

34.6 

35.6 

36.7 

1.0 

250 

21.9 

23.2  24.5 

25.71  27.0 

28.2 

29.5 

30.8 

32.0 

33.3 

34.5 

35.8 

37.1 

38.3 

1.3 

260 

20.3 

21  J  23.3 

24.9  26.4 

27.9 

29.4 

30.9 

32.4 

33.9 

35.4 

37.0 

38.5 

40.0 

1.5| 

270 

18.7 

20.,*;  22.2 

24.0  25.8 

27.5 

29.3 

31.1 

32.8 

34.6 

36.3 

38.1 

39.9  41.6 

1.8 

280  17.1 

19.1J  21.1  23.1 

25.2 

27.2 

29.2 

31.2 

33.2 

35.2 

37.3 

39.3 

41.3  43.3 

2.0 

290  15.5 

17.8  20.0,  22.3  24.6 

26.8 

29.1 

31.4 

33.6 

35.9 

38.2 

40.4 

42.7J  45.0 

2.3 

300  13.9 

16.4  18.9121.4  24.0 

26.5 

29.0 

31.5 

34.0 

36.6 

39.1 

41.6 

44.1 

46.6 

2.5  ! 

310  12.3 

15.0   17.8  20.6 

23.3 

26.1 

28.9 

31.7 

34.4 

37.2 

40.0 

42.7 

45.5 

48.3 

2.8  j 

320  10.7|l3.7   16.7;  19.7  22.7 

25.8 

28.8 

31.8 

34.8 

37.9 

40.9 

43.9 

46.9 

50.0 

3.0 

330 

9.0 

12.3 

15.6!  18.9;  22.1 

25.4 

28.7!  32.0 

35.2  38.5 

41.8 

45.1 

48.3 

51.6 

33 

340 

7.4)10.9 

14.5   18.0 

21.5 

25.1 

28.6  32.1   35.6 

39.2 

42.7 

46.2 

49.8 

53.3 

35 

350 

5.8 

9.6 

13.4:  17.1 

20.9 

24.7 

28.5  32.3  36.0 

39.8 

43.6 

47.4 

51.2 

54.9  3.8 

360 

4.2 

8.2 

12.3   16.3 

20.3 

24.4 

28.4  32.4 

36.4 

40.5 

44.5 

48.5 

52.6 

56.6 

4.0 

370 

2.6i   6.9 

11.1   15.4 

19.7 

24.01  28.3  32.6 

36.8 

41.1 

45.4 

49.7 

54.0 

58.3 

43 

380 

l.OJ  5.5 

10.0  14.6   19.1 

23.6J  28.2  32.7 

37.2 

41.8 

46.3  50.9 

55.4]  59.9 

4.5 

L_ 

0" 

50"  100" 

150"  200" 

1  :  

250"  300"  350" 

400"  450" 

500"!550" 

600"  650" 

— 

TABLE    LXXXI  I.     Moons  Horary  Motion  in  Latitude. 
Argument.     Arg.  II.  of  Latitude. 


05 

1.9 

11* 

111* 

IVs 

Vs 

0 

" 

" 

" 

" 

•  r 

ft 

0 

0 

9.3 

8.7 

7.1 

5.0 

2.9 

1.3 

30 

3 

9.3 

8.6 

6.9 

4.8 

2.7 

1.2 

27 

6 

9.2 

8.5 

6.7 

4.6 

2.5 

1.1 

24 

9 

9.2 

8.3 

6.5 

4.3 

2.3 

1.0 

21 

12 

9.2 

8.2 

6.3 

4.1 

2.1 

0.9 

18 

15 

9.1 

8.0 

6.1 

3.9 

2.0 

0.9 

15 

18 

9.1 

7.9 

5.9 

3.7 

1.8 

0.8 

12 

21 

9.0 

7.7 

5.7 

3.5 

1.7 

0.8 

9 

24 

89 

7.5 

5.4 

3.3 

1.5 

0.8 

6 

27 

88 

7.3 

5.2 

3.1 

1.4 

0.7 

.  3 

30 

87 

7.1 

5.0 

2.9 

1.3 

0.7 

0 

XI* 

Xo  ;  :,\,  iVM*:-  V!i* 

VI* 

M 


98  TABLE  LXXXIII. 

Moon's  Horary  Motion  in  Latitude. 

Arguments.  Preceding  equation,  and  Sum 
of  equations  of  Horary  Motion  in  Longi- 
tude, except  the  last  two. 


: 

Prec. 

H 

equ. 

0 

100 

200 

300 

400 

500 

600 

700 

0 

2.1 

1.8 

1.5 

1.2 

0.9 

0.6 

0.3 

0.0 

1 

.9 

1.6 

1.4 

1.1 

0.9 

0.7 

0.4 

0.2 

2 

.7 

1.5 

1.3 

1.1 

.0 

0.8 

0.6 

0.3 

3 

.5 

1.4 

1.2 

1.1 

.0 

0.9 

0.8 

0.6 

4 

.3 

1.2 

1.2 

1.1 

.1 

.0 

0.9 

0.9 

5 

.1 

1.1 

1.1 

1.1 

.1 

.1 

1.1 

1.1 

6 

0.9 

1.0 

1.0 

1.1 

.1 

.2 

1.3 

1.3 

7 

0.7 

0.8 

1.0 

1.1 

.2 

.3 

1.4 

1.6 

8 

0.5 

0.7 

0.9 

1.1 

.2 

.4 

1.6 

1.9 

9 

0.3 

0.6 

0.8 

1.1 

1.3 

.5 

1.8 

2.0 

10 

0.1 

0.4 

0.7 

1.0 

1.3 

1.6 

1.9 

2.2 

0 

100 

200 

300 

400 

500 

600 

700 

Constant  to  be  subtracted  237"  .2. 

TABLE  LXXXV. 

Moon's  Horary  Motion  in  Latitude. 

(Equations  of  second    order.) 
Arguments.     Preceding  equation,  and  Sum 
of  equations  of  Horary  Motion  in  Longi- 
tude, except  the  last  two. 


Prec 

" 

equ. 

0 

100 

200 

300 

400 

500 

600 

700 

0.00 

0.65 

0.57 

0.48 

0.39 

0.31 

0.21 

0.12 

0.00 

0.10  0.62 

055 

0.47 

0.39 

0.31  !o.23 

0.15 

0.04 

0.20  0.69 

0.53 

0.46 

0.39 

0.32  0.25 

0.18 

0.09 

0.30  0.66 

0.51 

0.45 

0.39 

0.33  0.27 

0.21 

0.13 

0.40  0.63 

0.48 

0.44 

0.39 

0.3410.29 

0.24 

0.17 

0.50  0.50  0.46 

0.43  0.38 

0.35 

0.30 

0.27 

0.21 

0.60  0.47  0.44 

0.42  0.38 

0.36 

0.32 

0.29 

0.25 

0.70  0.44(0.42 

0.40  0.38 

0.36 

0.34 

0.32 

0.30 

0.80 

0.41  ,0.40 

0.39 

0.38 

0.37 

0.36 

0.35 

0.34 

0.90 

0.38  0.38 

0.38 

0.38 

0.38 

0.38 

0.38 

0.38 

1.00 

0.35 

0.36 

0.37 

0.38 

0.39 

0.40 

0.41 

0.42 

1.10 

0.32 

0.34 

0.36 

0.38 

0.40 

0.42 

0.44 

0.46 

1.20 

0.29 

0.32 

0.34 

0.38 

0.40 

0.44 

0.47 

0.51 

1.30 

026 

0.30 

033 

0.38 

0.41 

0.46 

0.49 

0.55 

1.40 

0.23 

0.28 

0.32 

0.37 

0.42 

0.47 

0.52 

0.59 

1.50 

0.20 

0.25 

0.31 

037 

0.43 

0.49 

0.55 

0.63 

1.60 

0.17 

0.23 

0.30 

0.37 

0.44 

0.51 

0.58 

0.67 

1.70 

0.14 

0.21 

0.29 

0.37 

0.45 

0.53 

0.61 

0.72 

1.80 

0.11 

0.19 

0.28 

0.37 

0.45 

0.55 

0.64 

0.76 

0 

100 

200 

300 

400 

500 

600 

700 

TABLE  LXXXIV. 

Moorfs  Hor.  Motion  in  Lat 

(Equa.  of  second   order.) 

Argument.     Arg.  I  of  Lat. 


I 

I 

«  o 

» 

// 

»  ° 

0  0 

0.90 

0.90 

XII  0 

5 

0.83 

0.97 

25 

10 

0.75 

1.05 

20 

15 

0.68 

1.12 

15 

20 

0.61 

.19 

10 

25 

0.54 

.26 

5 

I   0 

0.47 

.33 

XI   0 

5 

0.41 

.39 

25 

10 

0.35 

.45 

20 

15 

0.29 

.51 

15 

20 

0.24 

.56 

10 

25 

0.20 

.60 

5 

II  0 

0.16 

.64 

X   0 

5 

0.12 

1.68 

25 

10 

0.09 

1.71 

20 

15 

0.07 

1.73 

15 

20 

0.05 

1.75 

10 

25 

0.04 

1.76 

5 

III  0 

0.04 

1.76 

IX   0 

5 

0.04 

1.76 

25 

10 

0.05 

1.75 

20 

15 

0.07 

1.73 

15 

20 

0.09 

1.71 

10 

25 

0.12 

1.68 

5 

rv  o 

0.16 

1.64 

VIII  0 

5 

0.20 

1.60 

25 

10 

0.24 

1.56 

20 

15 

0.29 

1.51 

15 

20 

0.35 

1.45 

10 

25 

0.41 

1.39 

5 

V  0 

0.47 

1.33 

VII   0 

5  ' 

0.54 

1.26 

25 

10 

0.61 

1.19 

20 

15 

0.68 

1.12 

15 

20 

0.75 

1.05 

10 

25 

0.83 

0.97 

5 

VI  0 

0.90 

0.90 

VI   0 

TABLE  LXXXVI. 
Mean  New  Moons  and  Arguments,  in  January. 


Years. 

Mean  New 
Moon  in. 
January. 

I. 

II. 

III. 

IV. 

N. 

d.  h  m. 

1821 

2  17  59 

0092 

7859 

80 

78 

823 

1822 

21  15  32 

0602 

7182 

78 

66 

930 

1823 

11  0  20 

0304 

5787 

61 

55 

953 

1824  B 

29  21  53 

0814 

5110 

59 

43 

060 

1825 

18  6  41 

0516 

3716 

42 

32 

083 

1826 

7  15  30 

0218 

2321 

25 

21 

105 

1827 

26  13  3 

0728 

1644 

24 

09 

213 

1828  B 

15  21  51 

0430 

0250 

07 

98 

235 

1829 

4  6  40 

0131 

8855 

90 

87 

257 

1830 

23  4  12 

0642 

8178 

88 

75 

365 

1831 

12  13  1 

0343 

6784 

71 

64 

387 

1832  B 

1  21  50 

0045 

5389 

54 

53 

409 

1833 

19  19  22 

0555 

4712 

53 

42 

517 

1834 

9  4  11 

0257 

3318 

36 

31 

539 

1835 

28  1  43 

0768 

2641 

34 

19 

647 

1836  B 

17  10  32 

0469 

1246 

17 

08 

669 

1837 

5  19  20 

0171 

9852 

00 

97 

692 

1838 

24  16  53 

0681 

9175 

99 

85 

799 

1839 

14  1  42 

0383 

7780 

82 

74 

822 

1840  B 

3  10  30 

0085 

6386 

65 

63 

844 

1841 

21  8  3 

0595 

5709 

63 

51 

951 

1842 

10  16  51 

0.297 

4314 

46 

40 

974 

i843 

29  14  24 

0807 

3637 

44 

28 

081 

1844  B 

18  23  13 

0509 

2243 

28 

17 

104 

1845 

7  8  1 

0211 

0848 

11 

06 

126 

1846 

26  5  34 

C721 

0171 

09 

94 

234 

1847 

15  14  22 

0423 

8777 

92 

84 

256 

i848B 

4  23  11 

0125 

7382 

75 

73 

278 

184-J 

22  20  43 

0635 

6705 

73 

61 

386 

1850 

12  5  32 

0337 

5311 

56 

50 

408 

1851 

1  14  21 

0038 

3916 

40 

39 

431 

1852  B 

20  11  53 

0549 

3239 

38 

27 

538 

1853 

8  20  42 

0251 

1845 

21 

16 

560 

1854 

27  18  14 

0761 

1168 

19 

04 

6G8 

1855 

17  3  3 

0463 

9773 

02 

93 

690 

1856  B 

6  11  51 

0164 

8379 

85 

82 

713 

1857 

24  9  24 

0675 

7702 

84 

70 

820 

1953 

13  18  13 

0376 

6307 

67 

59 

843 

1859 

3  3  1 

0078 

4913 

50 

48 

865 

18GOB 

22  0  34 

0588  |  4236 

48 

36 

972 

100 


TABLE  LXXXVII. 


Mean  Lunations  and  Changes  of  the  Arguments, 


Num 

Lunations. 

I. 

II. 

III. 

IV. 

N. 

d.   h   ni 

i 

14  18  22 

404 

5359 

58 

50 

43 

1 

29  12  44 

808 

717 

15 

99 

85 

2 

59  1  28 

1617 

1434 

31 

98 

170 

3 

88  14  12 

2425 

2151 

46 

97 

256 

4 

118  2  56 

3234 

2869 

61 

96 

341 

5 

147  15  40 

4042 

3586 

76 

95 

426 

6 

177  4  24 

4851 

4303 

92 

95 

511 

7 

206  17  8 

5659 

5020 

7 

94 

596 

8 

236  5  52 

6468 

5737 

22 

93 

682 

9 

265  18  36 

7276 

6454 

37 

92 

767 

10 

295  7  20 

8085 

7171 

53 

91 

852 

11 

324  20  5 

8893 

7889 

68 

90 

937 

12 

354  8  49 

9702 

8606 

83 

89 

22 

13 

383  21  33 

510 

9323 

98 

88 

108 

TABLE  LXXXVIII. 

Number  of  Days  from    the  commencement  of  the  year 
to  the  first  of  each  month- 


Months. 

Com. 

Bis.      j 

Days. 

Days. 

January 

0 

0 

February 

31 

31 

March  . 

59 

60 

April     . 

90 

91 

May      . 

120 

121 

June 

151 

152 

JuJy      . 

181 

182 

August  . 

212 

213 

September 
October     . 

243 
273 

244 
274 

November 

304 

305 

December 

334 

33f» 

TABLE  LXXXIX. 

Equations  for  New  and  Fall  Moon. 


101 


Arg. 

i  |  ii 

Arg. 

I 

II 

Arg!  Ill 

IV 

Arg 

h  in  1  h  in 

h  m 

h  m 

m 

m 

0 

4  20  I  10  10 

5000 

4  20 

10  10 

25 

3 

31 

25 

100 

4  36 

9  36 

5100 

4  5 

10  50 

26 

3 

31 

24 

200 

4  52 

9  2 

5200 

3  49 

11  30 

27 

3 

30 

23 

300  5  8 

8  28 

5300 

3  34 

12  9 

28 

3 

30 

22 

400 

5  24 

7  55 

5400 

3  19 

12  48 

29 

3 

30 

21 

500 

5  40 

7  22 

5500 

3  4 

13  26 

30 

3 

30 

20 

600  5  55 

6  49 

5600 

2  49 

14  3 

31 

3 

30 

19 

700  6  10 

6  17 

5700 

2  35 

14  39 

32 

4 

30 

18 

800  6  24 

5  46 

5800 

2  21 

15  13 

33 

4 

29 

17 

900  i  6  38 

5  15 

5900 

2  8 

15  46 

34 

4 

29 

16 

1000  '  6  51 

4  46 

6000 

1  55 

16  18 

35 

4 

29 

15 

1100  7  4 

4  17 

6100 

1  42 

16  48 

36 

5 

28 

14 

1200 

7  15 

3  50 

6200 

1  31 

17  16 

37 

5 

28 

13 

1300 

7  27 

3  24 

6300 

1  19 

17  42 

38 

5 

27 

12 

j  1400 

7  37 

2  59 

6400 

1  9 

18  6 

39 

5 

27 

11 

1500 

7  47 

2  35 

6500 

0  59 

18  28 

40 

6 

26 

10 

1600 

7  55 

2  14 

6600 

0  50 

18  48 

41 

6 

26 

9 

1700 

8  3 

1  53 

6700 

0  42 

19  6 

42 

7 

25 

8 

1800 

8  10 

1  35 

6800 

0  34 

19  21 

43 

7 

25 

7 

1900 

8  16 

1  18 

6900 

0  28 

19  33 

44 

7 

24 

6 

2000 

8  21 

1  3 

7000 

0  22 

19  44 

45 

8 

23 

5 

2100 

8  25 

0  51 

7100 

0  17 

19  52 

46 

8 

23 

4 

2200 

8  29 

0  40 

7200 

0  14 

19  57 

47 

9 

22 

3 

2300 

8  31 

0  32 

7300 

0  11 

20  0 

48 

9 

21 

2 

2400 

8  32 

0  25 

7400 

0  9 

20  1 

49 

10 

21 

1 

2500 

8  32 

0  21 

7500 

0  8 

19  59 

50 

10 

20 

0 

2600 

8  31 

0  19 

7600 

0  8 

19  55 

51 

10 

19 

99 

2700 

8  29 

0  20 

7700 

0  9 

19  48 

52 

11 

19 

98 

2800 

8  26 

0  23 

7800 

0  11 

19  40 

53 

11 

18 

97 

2900 

8  23 

0  28 

7900 

0  15 

19  29 

54 

12 

17 

96 

3000 

8  18 

0  36 

8000 

0  19 

19  17 

55 

12 

17 

95 

3100 

8  12 

0  47 

8100 

0  24 

19  2 

56 

13 

16 

94 

3200 

8  6 

0  59 

8200 

0  30 

18  45 

57 

13 

15 

93 

3300 

7  58 

1  14 

8300 

0  37 

18  27 

58 

13 

15 

92 

3400 

7  50 

1  32 

8400 

0  45 

18  6 

59 

14 

14 

91 

3500 

7  41 

1  52 

8500 

0  53 

17  45 

60 

14 

14 

90 

3600 

7  31 

2  14 

8600 

1  3 

17  21 

61 

15 

13 

89 

3700 

7  21 

2  38 

8700 

1  13 

16  56 

62 

15 

13 

88 

3800 

7  9 

3  4 

8800 

1  25 

16  30 

63 

15 

12 

87 

3900 

6  58 

3  32 

8900 

1  36 

16  3 

64 

15 

12 

86 

4000 

6  45 

4  2 

9000 

I  49 

15  34 

65 

16 

11 

85 

4100 

6  32 

4  34 

9100 

2  2 

15  5 

66 

16 

11 

84 

4200 

6  19 

5  7 

9200 

2  16 

14  34 

67 

16 

11 

83 

4300 

6  5 

5  41 

9300 

2  30 

14  3 

68 

16 

10 

82 

4400 

5  51 

6  17 

9400 

2  45 

13  31 

69 

17 

10 

81 

4500 

5  36 

6  54 

9500 

3  0 

12  58 

70 

17 

10 

80 

4600 

5  21 

7  32 

9600 

3  16 

12  25 

71 

17 

10 

79 

4700 

5  6 

8  11 

9700 

3  32 

11  52 

72 

17 

10 

78 

4800 

4  51 

8  50 

9SOO 

3  48 

11  18 

73 

17 

10 

77 

4900 

4  35 

9  30 

9900 

4  4 

10  44 

74 

17 

9 

76 

5000 

4  20 

10  10 

10000 

4  20 

10  10 

75 

17 

9 

75 

102 


TABLE  XC. 


Mean  Right  Ascensions  and  Declinations  of  50  principal  Fixed 
Stars,  for  the  beginning  of  1840. 


Stars'  Name. 

Mag 

light  Ascen. 

AnnualVar. 

Declination. 

Ann.  Var. 

ft     m      a 

8 

0       '           " 

~ 

1       Algenib 

2.3 

0     5    0.31 

+    3.0775 

14  17  38.82  N 

-1-  20.051 

2   /?  Andromedae 

2 

1     0  46.7 

3.309 

34  46  17.2   N 

19.35 

3      Polaris 

2.3 

1     2  10.38 

16.1962 

88  27  21.96N 

19.339 

4      Achernar 

1 

1  31  44.88 

2.2351 

58     3     5.13  S 

—  18.473 

5    a  Arietis 

3 

1  58     9.94 

3.3457 

22  42  11.81  N 

-r-  17.455 

f 

6    a  Ceti 

2.3 

2  53  55.34 

-r-    3.1257 

3  27  30.09  N 

+  14.561 

7    aPersei 

2.3 

3  12  55.97 

4.2280 

49  17     8.74N 

13.371 

8       AldebartM 

1 

4  26  44.77 

3.4264 

16  10  56.82  N 

7.949 

9       Capella 

1 

5     4  52.67 

4.4066 

45  49  42.81  N 

4.793 

10      Rig  el 

1 

5     6  51.09 

2.8783 

8  23  29.29  S 

—   4.620 

11    tfTauri 

2 

5  16  10.96 

4-    3.7820 

28  27  58.20  N 

-r-    3.825 

12    y  Orionis 

2 

5  16  33.1 

3.210 

6  11  55.3   N 

+    3.82 

13    a  Columbao 

2 

5  33  51.52 

21688 

34     9  47.41  S 

—   2.291 

14    a  Orionis 

1 

5  46  30.71 

3.2430 

7  22  17.14N 

+    1.191 

15       Canopus 

1 

6  20  24.18 

1.3278 

52  36  38.42  S 

1.778 

16      Sirnu 

1 

6  38     5.76 

+    2.6458 

16  30     4.79  S 

-r-    4.449 

17      Castor 

3 

7  24  23.06 

3.8572 

32  13  58.89N 

—   7.206 

18      Procyon 

1.2 

7  30  55.53 

3.1448 

5  37  48.92  N 

8.720 

19      PoZ/w* 

2 

,7  35  31.07 

3.6840 

28  24  25.57  N 

8.107 

20    aHydrae 

2 

9  19  43.57 

2.9500 

7  58     4.83  S 

+  15.341 

21      Regulus 

1 

9  59  50.93 

-1-    3.2220 

12  44  49.70  N 

—  17.356 

22    a  Ursae  Majoris 

1.2 

10  53  47.98 

3.8077 

62  36  48.93N 

19.221 

23   /JLeonis 

2.3 

11  40  53.69 

3.0660 

15  28     1.16N 

19.985 

24   0  Virginia 

34 

11  42  21.4 

3.124 

2  40     2  6   N 

19.98 

25    y  Ursae  Majoris 

2 

11  45  22.93 

3.1914 

54  35     4  67  N 

20.014 

26a»Crucis 

2 

12  17  43.7 

+    3.258 

62  12  47.  9S 

-f  19.99 

27       S/>fca 
28    0  Centauri 

1 
2 

13  16  46.36 
13  57  18.0 

3.1502 
3.491 

10  19  24.39  S 
35  34  41.9    S 

18.945 
17.499 

29    a  Draconis 

3.4 

14     0     2.8 

1.625 

65     8  32.1    N 

—  17.37 

30      Arcturvs 

1 

14     8  21.96 

2.7335 

20     1     7.67  N 

18.956 

31  a  2  Centauri 

1 

14  28  47.84 

-{-    4.0086 

60  10     6.24  S 

-f  15.  152 

32  a  2  Librae 

3 

14  42     2.44 

3.3088 

15  22  18.25S 

15.256 

33   />  Ursae  Minoris 

3 

14  51   14.66 

—   0.2787 

74  48  34.18N 

—  14.712 

34  y  a  Ursae  Minoris 

3.4 

15  21     1.3 

—   0.179 

72  24  14.1    N 

12.81 

35    a  Coronae  Borealis 

2 

15  27  54.87 

+    2.5277 

27  15  27.71  N 

12.361 

36    a  Serpentis 

2.3 

15  36  2343 

+    2.9386 

6  56     2.  SON 

—  11.770 

37  /JScorpii 

2 

15  56     8.68 

3.4729 

19  21  38.82  S 

-1-  10.330  i 

38      Ant&res 

1 

16  19  36.49 

3.6625 

26     4  13.13S 

8.519 

J9    a  Herculis 

3.4 

17     7  21.30 

2.7317 

14  34  41.43  N 

—   4.576 

40    uOphiuchi 

2 

17  27  30  56 

2.7724 

12  40  58.65  N 

2.844 

41    fi  Ursae  Minoris 

3 

18  23  56.48 

—  19.2072 

86  35  28.89  N 

+    2.161 

42       Vega 

1 

18  31  31.19 

+    2.0116 

38  38  16.85  N 

2.742 

43      Altair 

1 

19  42  58.61 

2.9255 

8  27     0.21  N 

8.701 

44  a  2  Capricomi 

3 

20     9  10.34 

3.3323 

13     2     5.57  S 

—  10.705 

45    a  Cygni 

1 

20  35  58.80 

2.0416 

44  42  41.38N 

+  12.614 

46    a  Aquarii 

3 

21  57  33.93 

+    3.0835 

1     5  38.00  S 

—  17.256 

47      Fomalhaut 

1 

22  48  47.67 

3.3114 

30  28     4.91  S 

19.092 

48   /?Pegasi 

2 

22  56     1.1 

2.878 

27  13     1.7   N 

-f  19.255 

49      Markal 

2 

22  56  47.75 

2.9771 

14  20  46.92N 

19295 

50    a  Andromedae 

1 

24     0     7.72 

3.0704 

28  12  27.06  N 

20.056  i 

TABLE  XCI. 


103 


Constants  for  the  Aberration  and  Nutation  in  Right  Ascension 
and  Declination  of  the  Stars  in  the  preceding  Catalogue 


\ 

Aberration. 

Nutation. 

§ 

M       ! 

t 

N 

¥ 

M' 

6' 

N' 

*    °    ' 

«    °     ' 

8      °        ' 

8       0        ' 

1 

8  28  47 

0.1087 

7  27  12,0.9657 

6     8   24 

0.0300 

5  28  30 

0.8381 

2 

8  13  39 

0.1830 

6   19   12    1.0740 

6  19   53 

0.0838 

5   10     8 

0.8496 

3 

8  13  51 

1.6526 

5  16  57    1.3052 

8  16      7    1.3427 

5   10  22 

0.8493 

4 

8     5  20,0.3801 

10  26  46    1.2798 

4  10    12 

0.0775 

5     0  31 

0.8629 

5 

7  28  26 

0.1397 

702 

0.8972 

6  11      1 

0.0695 

4  22  53 

0.8765 

6 

7  14  11 

0.1149 

8  23     8 

0.8678 

6     1   26 

0.0322 

4    8   16 

0.9078 

7 

7     9  30 

0.3020 

535 

1.0630 

6  18    13 

0.1849 

4     3  47 

0.9179 

8 

6  21  43 

0.1447 

7  23  12 

0.5760 

6     3   27 

0.0726 

3  17  54 

0.9502 

9 

6   12  51 

0.2875 

3  25  37 

0.9112 

6     5  46 

0.1830 

3  10  29 

0.9605 

10 

6   12  20 

0.1355 

9     3  42 

1.0300 

5  28  47 

1.9966 

3  10     4 

0.9608 

11 

6  10  13 

0.1873 

4  19  21 

0.3917 

6     2  52 

0.1008 

3     8   19 

0.9626 

12 

6  10     6 

0.1340 

8  26     4 

0.7851 

6     0  40 

0.0441 

3     8   14 

0.9626 

13 

665 

0.2145 

9     4  24 

1.2348 

5  26   18 

1.8750 

3     4  57 

0.9648 

14 

6     3  13 

0.1361 

8  28  23 

0.7521 

6     0   15 

0.0481 

3     2  37 

0.9657 

15 

5  25  22 

0.3491 

8  25  53 

1.2960 

6     8  46 

1.6679 

2  26   15 

0.9657 

16 

5  21  21 

0.1501 

8  25  51 

1.1152 

6     1   51 

1.9658 

2  22  58 

0.9636 

17 

5   10  40 

0.2010 

1     2   17 

0.6620 

5  24     2 

0.1257 

2   14     6 

0.9535 

18 

596 

0.1297 

9     6  54 

0.8071 

5  28  47 

0.0414 

2   12  47 

0.9513 

19 

582 

0.1829 

0  14  32 

0.6052 

5  24     2 

0.1114 

2   11   53 

0.9499 

20 

4  12  39 

0.1158 

8   17  31 

0.9967 

6     3  41 

0.0081 

1    18  37 

0.9007 

21 

4     2  22 

0.1162 

10     3  47 

0.8457 

5  23  47 

0.0480 

1     7  59 

0.8782 

22 

3  18     7 

0.4366 

0     3  28 

1.2394 

4  18  58 

0.2407 

0  21   57 

0.8520 

23 

3     5  21 

0.1117 

10     6  20 

0.9621 

5  20  56 

0.0344 

0     6  35 

0.8393 

24 

3     4  57 

0.0958 

9     6  51 

0.9075 

5  28  25 

0.0253 

065 

0.8390 

25 

348 

0.3229 

11   17  28 

1.2298 

4  21   46 

0.1465 

055 

0.8383 

26 

2  25  19 

0.4261 

685 

1.2585 

7  16     2 

0.2089 

11   24  14 

0.8390 

27 

2     9  22 

0.1066 

8     3  31 

0.8862 

6     5  51 

0.0154    11      56 

0.8559 

28 

1  28  40 

0.1942 

6     7  12 

1.0176 

6  17  31 

0.1062    10   23     8  i  0.8760 

29 

1  27  53 

0.4824 

10  23  28 

1.2995 

3  25  50 

0.1090  '10   22   16    0.8777 

30 

1  25  46 

0.1336 

9  28   18 

1.0974 

5  18  49 

1.99371  10  20     1    0.8822 

31 

1  20  32 

0.4123 

5     7  54 

1.1820 

6  29     6 

0.2460  il  10   14  36    0.8937 

32 

I  17  26 

0.1273 

7  18  24 

0.6923 

6     6  29    0.0593J  10   11   28  !  0.9006 

33 

i  14  42 

0.6961 

10  15     5 

1.3087 

2  26  45 

0.  2235  j  10     8  47   0.9066 

34 

1     7  20   0.6386 

10     7  33 

1.3087 

2  27     7 

0.0960 

10      1   45  '  0.9225 

35 

1     5  45   0.1704 

9  22  28 

1.1785 

5   17  18 

1.9510 

10     0   18 

0.9257 

36 

1     3  43   0.1237 

9     8  22 

0.9994 

5  27  30 

0.0058 

9   28   26 

0.9298 

37 

0  28  58   0.1485 

744 

0.6237 

6     5  20 

0.0795      9  24   12)0.9386 

38 

0  23  24   0.1728 

5  27  59 

0.5816 

6     5  49 

0.1029      9    j9  21    0.9478 

39 

0  12  13   0.1451 

9     5  25 

1.0962 

5  27  45 

1.9742      9     9  58    0.9610 

40 

0     7  34   0.1427 

934 

1.0786 

5  28  48 

1.9803 

969 

0.9642 

41 

11  23  47   1.3571 

8  22  49 

1.2821 

11   19  31 

0-8257 

8  24  57 

0.9650 

42 

11  22  50   0.2393 

8  24  29 

1.2545 

6     5  31 

1.8436  i    8  24  10    0.9644 

43 

11     6   15   0.1309 

8  22  59 

1.0237 

0     2  16 

1.9988 

8   10  21    0.9472 

44 

11     0     2   0.1341 

9  29  33 

0.6961 

5  26  12 

0061)9 

8     4  55   0.9368 

45 

10  23  29   0.2668 

8     0  39 

1.2634 

6  28  32 

1.9042 

7  29     0   0.9242 

46 

10    2  57  0.1057 

9     2  31 

0.8988 

5  29  S6 

0.0264 

7     8  37   0.8794 

47 

9  19  26   0.1638 

11     7  34    1.0271 

5  13     8 

0.0765 

6  23  30  '0.8540 

48 

9  17  29   0.1491 

7  17     0    1.1171 

6  17     2 

0.0162 

6  21    13!  0.8511 

49 

9  17  17 

0.1120 

825 

1.0138 

6     8  23 

0.0157 

6  20  58    0.8508 

50 

906 

0.1495 

7    6  42 

1.0785 

6  17  20 

0.0444 

6     0     8  1  0.8380 

104 


TABLE  XCII. 


Mean  Longitudes  and  Latitudes  of  some  of  the  principal  Fixed 
Stars  for  the  beginning  of  1 840,  with  their  Annual  Variations. 


Stars'  Name. 

Mag 

Longitude. 

Annual 
Var. 

Latitude. 

Annual 
Var. 

a  Arietis 

3 

g           0      '          " 

1     5  25  27.6 

50.277 

0      '            " 

9  57  40.9  N 

+  0.161 

Aldebaran 

1 

2     7  33     5.9 

50.210 

5  28  38.0  s 

—  0.335 

Capella 

1 

2  19  37  17.8 

50.302 

22  51  44.4  N 

—  0.052 

Polaris 

2.3 

2  26  19  20.1 

47.959 

66     4  59.  5  N 

+  0.552 

Sirius 

1 

3  11  52  32.9 

49.488 

39  34    4.3  S 

+  0.319 

Canopus 

1 

3  12  44  59.6 

49.366 

75  50  57.6  S 

+  0.459 

Pollux 

2 

3  21     0  22.0 

49.502 

6  40  20.2  N 

+  0.255 

Regulus 

1 

4  27  36   13.2 

49.946 

0  27  38.  3  N 

+  0.220 

Spica 

1 

6  21  36  29.2 

50.085 

2     2  29.7  S 

+  0.171 

Arcturus 

1 

6  22     0     4.7 

50.711 

30  51   17.5  N 

+  0.214 

Antares 

1 

8     7  31  45.2 

50.120 

4  32  51.  6  S 

4  0.424 

Altair 

1.2 

9  29  31     5.9 

50.795 

29  18  37.3  N 

4-  0.080 

Fomalhaut 

1 

11     1  36  22.0 

50.595 

21     6  49.  7  S 

+  0.213 

Achernar 

1 

11   13     2     5.3 

50.346 

17     6  17.3  S 

—  0.083 

a  Pegasi 

2 

11  21   15  24.7 

50.112 

19  24  40.9  N 

+  0.098 

TABLE  added  to  TABLE  XC. 

Mean  Right  Ascensions  and  Declinations  of  Polaris  and  <*  Ursae 
Minoris  for  1S30,  1840,  1850,  and  1860. 


Stars. 

Years 

Right  Asc. 

Ann.  Var. 

Declination. 

Ann.  Var. 

O          '            '• 

// 

0         /             " 

„ 

1830 

0  59  30.76 

+  15.478 

88  24     8.82 

+  19.371 

Polaris 

1840 
1850 

1     2  10.32 
1     5     0.29 

16.470 
17.567 

88  27  22.43 
88  30  35.40 

19.309 
19.240 

1860 

1     8     1.79 

18.784 

88  33  47.64 

19.163 

1830 

18  27     5.13 

—  19.167 

86  35     5.70 

+    2.363 

i  Ursae  Minoris 

1840 

18  23  53.03 

19.241 

86  35  27.93 

2.085 

1850 

18  20  40.21 

19.305 

86  35  47.36 

1.805 

1860 

18  17  26.77 

19.360 

86  36     3  97 

1.523 

TABLE  XCIII. 

Second  Differences. 


105 


Hours  &  Minutes. 

1' 

2' 

3' 

4' 

5' 

6' 

7' 

8' 

9' 

10' 

ir 

h  m 

h  m 

!  0  0 

12  0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0  10 

11  50 

0.4 

0.8 

1.2 

1.6 

2.0 

2.4 

2.9 

3.3 

3.7 

4.1 

4.5 

0  20 

11  40 

0.8 

1.6 

2.4 

3.2 

4.1 

4.9 

5.7 

6.5 

7.3 

81 

8.9 

0  30 

11  30 

1.2 

2.4 

3.6 

4.8 

6.0 

7.2 

8.4 

9.6 

10.8 

12.0 

13.2 

0  40 

11  20 

1.6 

3.1 

4.7 

6.3 

7.9 

9.4 

11.0 

12.6 

14.2 

15.7 

17.3 

0  50 

11  10 

1.9 

3.9 

5.8 

7.8 

9.7 

11.6 

13.6 

15.5 

17.4 

19.4 

21.4 

1  0 

11  0 

2.3 

4.6 

6.9 

9.2 

11.5 

13.8 

16.0 

18.3 

20.6 

22.9 

25.2 

1  10 

10  50 

2.6 

5.3 

7.9 

10.5 

13.2 

15.8 

18.4 

21.1 

23.7 

26.3 

29.0 

1  20 

10  40 

3.0 

5.9 

8.9 

11.9 

14.8 

17.8 

20.7 

23.7 

26.7 

29.6 

32.6 

1  30 

10  30 

3.3 

6.6 

9.8 

13.1 

16.4 

19.7 

23.0 

26.3 

29.5 

32.8 

36.1 

1  40 

10  20 

3.6 

7.2 

10.8 

14.4 

17.9  21.5 

25.1 

28.7 

32.3 

35.9 

39.5 

1  50 

10  10 

3.9 

7.8 

11.6 

15.5 

19.4 

23.3 

27.2 

31.0 

34.9 

38.8 

42.7 

2  0 

10  0 

4.2 

8.3 

12.5 

16.7 

20.8 

25.0 

29.2 

33.3 

37.5 

41.7 

45.8 

2  10 

9  50 

4.4 

8.9 

13.3 

17.8 

22.2  26.6 

31.1 

35.5 

40.0 

44.4 

48.8 

2  20 

9  40 

4.7 

9.4 

14.1 

18.8 

23.5 

28.2 

32.9 

37.6 

42.3 

47.0 

51.7 

2  30 

9  30 

4.9 

9.9 

14.8 

19.8 

24.7 

29.7 

34.6 

39.6 

44.5 

49.5 

54.4 

2  40 

9  20 

5.2 

10.4 

15.6 

20.7 

25.9 

31.1 

36.3 

41.5 

46.7 

51.9 

57.0 

2  50 

9  10 

5.4 

10.8 

16.2 

21.6 

27.1 

32.5 

37.9 

43.3 

48.7 

54.1 

59.5 

3  0 

9  0 

5.6 

11.3 

16.9 

22.5 

28.1 

33.8 

39.4 

45.0 

50.6 

56.3 

61.9 

3  10 

8  50 

5.8 

11.7 

17.5 

23.3 

29.1 

35.0 

40.8 

46.6 

52.4 

58.3 

64.1 

3  20 

8  40 

6.0 

12.0 

18.1 

24.1 

30.1 

36.1 

42.1 

48.1 

54.2 

60.2 

66.2 

3  30 

8  30 

6.2 

12.4 

18.6 

24.8 

31.0 

37.2 

43.4 

49.6 

55.8 

62.0 

68.2 

3  40 

8  20 

6.4 

12.7 

19.1 

25.5 

31.8 

38.2 

44.6 

50.9 

57.3 

63.7 

70.0 

3  50 

8  10 

6.5 

13.0 

19.6 

26.1 

32.6 

39.1 

45.7 

52.2 

58.7 

65.2 

71.7 

4  0 

8  0 

6.7 

13.3 

20.0 

26.7 

33.3 

40.0 

46.7 

53.3 

60.0 

66.7 

73.3 

4  10 

7  50 

6.8 

13.6 

20.4 

27.2 

34.0 

40.8 

47.6 

54.4 

61.2 

68.0 

74.8 

4  20 

7  40 

6.9 

13.8 

20.8 

27.7 

34.6 

41.5 

48.4 

55.4 

62.3 

69.2 

76.1 

4  30 

7  30 

7.0 

14.1 

21.1 

28.1 

35.2  42.2 

49.2 

56.2 

63.3 

70.3 

77.3 

4  40 

7  20 

7.1 

14.3 

21.4 

28.5 

35.6 

42.8 

49.9 

57.0  64.2 

71.3 

78.4 

4  50 

7  10 

7.2 

14.4 

21.6 

28.9 

36.1 

43.3 

50.5 

57.7 

64.9 

72.2 

79.4 

5  0 

7  0 

7.3 

14.6 

21.9 

29.2 

36.5 

43.8 

51.0 

58.3 

65.6 

72.9 

80.2 

5  10 

6  50 

7.4 

14.7 

22.1 

29.4 

36.8 

44.1 

51.5 

58.8  66.2 

73.6 

80.9 

5  20 

6  40 

7.4 

14.8 

22.2 

29.6 

37.0 

44.4 

51.9 

59.3  66.7 

74.1 

81.5 

5  30 

6  30 

7.4 

14.9 

22.3 

29.8 

37.2 

44.7 

52.1 

59.6  67.0 

74.5 

81.9 

5  40 

6  20 

7.5 

15.0 

22.4 

29.9 

37.4 

44.9 

52.3 

59.8  67.3 

74.8 

82.2 

5  50 

6  10 

7.5 

15.0 

22.5 

30.0 

37.5 

45.0 

52.5 

60.0  67.4  74.9 

82.4 

6  0 

6  0 

7.5 

15.0 

22.5 

30.0 

37.5 

45.0 

52.5 

60.0  67.5 

75.0 

82.5 

N 


106 


TABLE  XCIII. 

Second  Differences. 


Hours  &  Mm.  10" 

20" 

30"  40" 

50" 

1" 

2" 

3" 

4'' 

5" 

6" 

7" 

8" 

9" 

h  m 

h  m  " 

•' 

" 

0  0 

12  0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0  10 

11  50 

0.1 

0.1 

0.2 

0.3 

0.3 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1  0.1 

020 

11  40 

0.1 

0.3 

0.4 

0.5 

0.7 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1  j  0.1 

0  30 

11  30 

0.2  0.4 

0.6 

0.8 

1.0 

0.0 

0.0 

0.1 

0.1 

O.I 

0.1 

0.1 

0.2  0.2 

0  40  11  20 

0.3 

0.5 

0.8 

1.0 

1.3 

0.0 

0. 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2  0.2 

0  50  11  10 

0.3 

0.6 

1.0 

1.3 

1.6 

0.0 

0. 

0.1 

0.1 

0.2 

0.2 

0.2 

0.3  0.3 

1  0 

11  0 

0.4 

0.8 

1.1 

1.5 

1.9 

0.0 

0. 

0.1 

0.2 

0.2 

0.2 

0.3 

0.3  0.3 

10 

10  50 

0.4 

0.9  1.3 

1.8 

2.2 

0.0 

0. 

0.1 

0.2 

0.2 

0.3 

0.3 

0.4  0.4 

20 

10  40 

0.5 

1.0  1.5 

2.0 

2.5 

0.0 

0. 

0.1 

0.2 

0.2 

0.3 

0.3 

0.4  0.4l 

30 

10  30 

0.5 

1.1  1.6 

2.2 

2.7 

0.1 

0.1 

0.2 

0.2 

0.3 

0.3 

0.4 

0.4 

0.5 

40 

10  20 

0.6 

1.2 

1.8 

2.4 

3.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.4 

0.5 

0.5 

50 

10  10 

0.6 

1.3 

1.9 

2.6 

3.2 

0.1 

0.1 

0.2 

0.3 

0.3 

0.4 

0.5 

0.5 

0.6 

2  0 

10  0 

0.7  1.4 

2.1 

2.8 

3.5 

0.1 

0.1  0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.6 

2  10 

9  50 

0.7 

1.5 

2.2 

3.0 

3.7 

0.1 

0.1 

0.2 

0.3 

0.4 

0.4 

0.5 

0.6 

0.7 

2  20 

9  40 

0.8 

1.6 

2.3 

3.1 

3.9 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.5 

0.6 

0.7 

2  30 

9  30 

0.8 

1.6 

2.5 

3.3 

4.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.7 

2  40 

9  20 

0.9 

1.7 

2.6|3.5 

4.3 

0.1 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

2  50 

9  10 

0.9 

1.8 

2.7  !  3.6 

4.5 

0.1 

0.2 

0.3 

0.4  0.5 

0.5 

0.6 

0.7 

0.8 

3  0 

9  0 

0.9 

1.9  2.8 

3.8 

4.7 

0.1 

0.2 

0.3 

0.4  0.5 

0.6 

0.7 

0.7 

0.8 

3  10 

8  50 

1.0  1.912.9 

3.9 

4.9 

0.1 

0.2 

0.3 

0.4  0.5 

0.6 

0.7 

0.8 

0.9 

3  20 

8  40 

1.0  2.0  3.0 

4.0 

5.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

3  30 

8  30 

1.0  2.1  3.1 

4.1 

5.2 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

3  40 

8  20 

1.1  2.1  i3.2 

4.2  5.3 

0.1 

0.2 

0.3 

0.4  |  0.5 

0.6 

0.7 

0.8 

1.0 

3  50 

8  10 

1.1  2.2 

3.3 

4.3  5.4 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

0.9 

1.0 

4  0 

8  0 

1.1  2.2 

3.3 

4.4  5.6 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.8 

0.9 

1.0 

4  10 

7  50 

1.1  2.3 

3.4 

4.5 

5.7 

0. 

0.2 

0.3 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

4  20 

7  40 

1.2 

2.3 

3.5 

4.6 

5.8 

0. 

0.2 

0.3 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

4  30 

7  30 

1.2 

2.313.5 

4.7 

5.9 

0. 

0.2 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1.1 

4  40 

7  20 

1.2  2.4  j  3.6 

4.8 

5.9 

0. 

0.2 

0.4 

0.5 

0.6 

0.7 

0.8 

1.0 

1.1 

4  50 

7  10 

1.2 

2.4 

3.6 

4.8 

6.0 

0. 

0.2 

0.4 

0.5 

0.6 

0.7 

0.8 

1.0 

1.1 

5  0 

7  0 

1.2 

2.4 

3.6 

4.9 

6.1 

0. 

0.2 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1.1 

5  10 

6  50 

1.2 

2.5 

3.7 

4.9 

6.1 

0. 

0.2 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1.1 

5  20 

6  40 

1.2 

2.5 

3.7 

4.9 

6.1 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1.1 

5  30 

6  30 

1.2 

2.5 

3.7 

5.0 

6.2 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.9 

LO 

1.1 

5  40 

6  20 

1.2 

2.5 

3.7 

5.0 

6.2 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1.1 

5  50 

6  10 

1.2 

2.5 

3.7 

5.0 

6.2 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1.1 

6  0 

6  0 

1.3 

2.6 

3.8 

5.0 

6.3 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1.1 

TABLE  XCIV. 


107 


Third  Differences. 


Time  after 

Time  after 

noon  or 

10" 

20" 

30" 

40" 

50" 

1' 

2' 

3' 

4' 

5' 

noon  or 

midnight. 

midnight. 

Oh.  Om. 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

12h.   Om. 

0     30 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.4 

0.5 

0.7 

0.9 

11     30 

1       0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.3 

0.6 

1.0 

1.3 

1.5 

11       0 

1     30 

0.1 

0.1 

0.2 

0.3 

0.3 

0.4 

0.8 

1.2 

1.6 

2.1 

10     30 

2       0 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.9 

1.4 

1.9 

2.3 

10       0 

2     30 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

1.0 

1.4 

1.9 

2.4 

9     30 

3       0 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.9 

1.4 

1.9 

2.3 

9       0 

3     30 

0.1 

0.1 

0.2 

0.3 

0.4 

0.4 

0.9 

1.3 

1.7 

2.2 

8     30 

4      0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.7 

1.1 

1.5 

1.9 

8      0 

4    30 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.6 

0.9 

1.2 

1.5 

7    30 

5      0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.4 

0.6 

oa 

1.0 

7      0 

5    30 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

6     30 

6      0 

.  J 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

6       0 

TABLE  XCV. 

Fourth  Differences. 


Time  after 
noon  or 

midnight. 

10" 

20" 

30" 

40" 

50" 

1- 

2' 

3' 

Time  after 
noon  or 
midnight. 

h.     m. 

h.     m. 

0       0 

0.0 

0.0 

0.0 

00 

0.0 

0.0 

0.0 

0.0 

12      0 

0     30 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.4 

0.6 

11     30 

1       0 

0.1 

0.1 

0.2 

0.3 

0.3 

0.4 

0.8 

1.2 

11       0 

1     30 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

1.2 

1.7 

10    30 

2      0 

0.1 

0.2 

04 

0.5 

0.6 

0.7 

1.5 

2.2 

10      0 

2     30 

0.1 

0.3 

0.4 

0.6 

0.7 

0.9 

1.8 

2.7 

9    30 

3       0 

0.2 

0.3 

0.5 

0.7 

0.9 

1.0 

2.1 

3.1 

9      0 

3    30 

0.2 

0.4 

0.6 

0.8 

0.9 

1.1 

2.3 

3.4 

8    30 

4      0 

0.2 

0.4 

0.6 

0.8 

1.0 

1.2 

2.5 

3.7 

8      0 

4    30 

0.2 

0.4 

0.7 

0.9 

1.1 

1.3 

2.6 

3.9 

7    30 

5      0 

0.2 

0.5 

0.7 

0.9 

1.1 

1.4 

2.7 

4.1 

7      0 

5    30 

0.2 

0.5 

0.7 

0.9 

1.2 

1.4 

2.8 

4.2 

6    30 

6      0 

0.2 

0.5 

0.7 

0.9 

1.2 

1.4 

2.8 

4.2 

6      0 

108 


TABLE  XCVI.     Logistical  Logarithms. 


f 

0 
0 

1     2 

3 
180 

4 

5     6 

7 

8 

9 

" 

60  !  120 

240 

300 

360 

420 

480 

540 

0 

1.7782 

1.4771 

1.3010 

1.1761 

1.0792 

1.0000 

9331 

8751 

8239 

1 

3.5563 

1.7710 

1.4735 

1.2986 

1.1743 

1.0777 

9988 

9320 

8742 

8231 

2 

3.2553 

1.7639 

1.4699 

1.2962 

L1725 

1.0763 

9976 

9310 

8733 

8223 

3 

3.0792 

1.7570 

1.4664 

1.2939 

1.1707 

1.0749 

9964 

9300 

8724 

8215 

4 

2.9542 

1.7501 

1.4629 

1.2915 

1.1689 

1.0734 

9952 

9289 

8715 

8207 

5 

2.8573 

1.7434 

1.4594 

1.2891 

1.1671 

1.0720 

9940 

9279 

8706 

8199 

6 

2.7782 

1.7368 

1.4559 

1.2868 

1.1654 

1.0706 

9928 

9269 

8697 

8191 

7 

2.7112 

1.7302 

1.4525 

1.2845 

1.1636 

1.0692 

9916 

9259 

8688 

8183 

8 

2.6532 

1.7238 

1.4491 

1.2821 

1.1619 

1.0678 

9905 

9249 

8679 

8175 

9 

2.6021 

1.7175 

1.4457 

1.2798 

1.1601 

1.0663 

9893 

9238 

8670 

8167 

10 

2.5563 

1.7112 

1.4424 

12775 

1.1584 

1.0649 

9881 

9228 

8661 

8159 

11 

2.5149 

1.7050 

1.4390 

1.2753 

1.1566 

1.0635 

9869 

9218 

8652 

8152 

12 

2.4771 

1.6990 

1.4357 

1  2730 

1.1549 

1.0621 

9858 

9208 

8643 

8144 

13 

2.4424 

1.6930 

1.4325 

1.2707 

1.1532 

1.0608 

9846 

9198 

8635 

8136 

14 

2.4102 

1.6871 

1.4292 

1.2685 

1.1515 

1.0594 

9834 

9188 

8626 

8128 

15 

2.3802 

1.6812 

1.4260 

1.2663 

1.1498 

1.0580 

9823 

9178 

8617 

8120 

16 

2.3522 

1.6755 

1.4228 

1.2640 

1.1481 

1.0566 

9811 

9168 

8608 

8112 

17 

2.3259 

1.6698 

1.4196 

1.2618 

1.1464 

1.0552 

9800 

9158 

8599 

8104 

18 

2.3010 

1.6642 

1.4165 

1.2596 

1.1447 

1.0539 

9788 

9148 

8591 

8097 

19 

2.2775 

1.6587 

1.4133 

1.2574 

1.1430 

1.0525 

9777 

9138 

8582 

8089 

20 

2.2553 

1.6532 

1.4102 

1.2553 

1.1413 

1.0512 

9765 

9128 

8573 

8081 

21 

22341 

1.6478 

1.4071 

1.2531 

1.1397 

1.0498 

9754 

9119 

8565 

8073 

22 

22139 

1.6425 

1.4040 

1.2510 

1.1380 

1.0484 

9742 

9109 

8556 

8066 

23 

2.1946 

1.6372 

1.4010 

1.2488 

1.1363 

1.0471 

9731 

9099 

8547 

8058 

24 

2.1761 

1.6320 

1.3979 

1.2467 

1.1347 

1.0458 

9720 

9089 

8539 

8050 

25 

2.1584 

1.6269 

1.3949 

1.2445 

1.1331 

1.0444 

9708 

9079 

8530 

8043 

26 

2.1413 

1.6218 

1.3919 

1.2424 

1.1314 

1.0431 

9697 

9070 

8522 

8035 

27 

2.1249 

1.6168 

1.3890 

1.2403 

1.1298 

1.0418 

9686 

9060 

8513 

8027 

28 

2.1091 

1.6118 

1.3860 

1.2382 

1.1282 

1.0404 

9675 

9050 

8504 

8020 

29 

2.0939 

1.6069 

1.3831 

1.2362 

1.1266 

1.0391 

9664 

9041 

8496 

8012 

30 

2.0792 

1.6021 

1.3802 

1.2341 

1.1249  1.0378 

9652 

9031 

8487 

8004 

31 

2.0649 

1.5973 

1.3773 

1.2320 

1.1233 

1.0365 

9641 

9021 

8479 

7997 

32 

2.0512 

1.5925 

1.3745 

1.2300 

1.1217 

1.0352 

9630 

9012 

8470 

7989 

33 

2.0378 

1.5878 

1.3716 

1.2279 

1.1201 

1.0339 

9619 

9002 

8462 

7981 

34 

2.0248 

1.5832 

1.3688 

1.2259 

1.1186 

1.0326 

9608 

8992 

8453 

7974 

35 

2.0122 

1.5786 

1.3660 

1.2239 

1.1170 

1.0313 

9597 

8983 

8445 

7966 

36 

2.0000 

1.5740 

1.3632 

1.2218 

1.1154 

1.0300 

9586 

8973 

8437 

7959 

37 

1.9881 

1.5695 

1.3604 

1.2198 

1.1138 

1.0287 

9575 

8964 

8428 

7951 

38 

1.9765 

1.5651 

1.3576 

1.2178 

1.1123 

1.0274 

9564 

8954 

8420 

7944 

33 

1.9652 

1.5607 

1.3549 

1.2159 

1.1107 

1.0261 

9553 

8945 

8411 

7936 

40 

1.9542 

1.5563 

1.3522 

1.2139 

1.1091 

1.0248 

9542 

8935 

8403 

7929 

41 

1.9435 

1.5520 

1.3495 

1.2119 

1.1076 

1.0235 

9532 

8926 

8395 

'7921 

42 

1.9331 

1.5477 

1.3468 

1.2099 

1.1061  !  1.0223 

9521 

8917 

8386 

7914 

43 

1.9228 

1.5435 

1.3441 

1.2080 

1.1045 

1.0210 

9510 

8907 

8378 

7906 

44 

1.9128 

1.5393 

1.3415 

1.2061 

1.1030 

1.0197 

9499 

8898 

8370 

7899 

45 

1.9031 

1.5351 

1.3388 

1.2041 

1.1015 

1.0185 

9488 

8888 

8361 

7891 

46 

1.8935 

1.5310 

1.3362 

1.2022 

1.0999 

1.0172 

9478 

8879 

8353 

7884 

47 

1.8842 

1.5269 

1.3336 

1.2003 

1.0984 

1.0160 

9467 

8870 

8345 

7877 

48 

1.8751 

1.5229 

1.3310 

1.1984 

1.0969 

1.0147 

9456 

8861 

8337 

7869 

49 

1.8661 

1.5189 

1.3284 

1.1965 

1.0954 

1.0135 

9446 

8851 

8328 

7862 

50 

1.8573 

1.5149 

1.3259 

1.1946 

1.0939  1.0122 

9435 

8842 

8320 

7855 

51 

1.8487 

1.5110]  1.3233 

1.1927 

1.0924  1.0110 

9425 

8833 

8312 

7847 

52 

1.8403 

1.5071 

1.3208 

1.1908 

1.0909  1.0098 

9414 

8824 

8304 

7840 

53 

1.8320  1.5032 

1.3183 

1.1889  1.0894  1.0085   9404 

8814 

8296 

7832 

54 

1.8239  !  1.4994 

1.3158 

1.1871  1.0880  1.0073'  9393 

8805 

8288 

7825 

55 

1.  81  59  j  1.4956  1.3133 

1.1852 

1.0865 

1.0061 

9383 

8796  8279 

7818 

56 

1.8081 

1.4918  1.3108 

1.1834 

1.0850 

1.0049 

9372 

8787 

8271 

7811 

67 

1.8004 

1.4881 

1.3083 

1.1816 

1.0835 

1.0036 

9362 

8778 

8263 

7803 

58 

1.7929  1.4844 

1.3059 

1.1797 

1.0821  1.0024 

9351 

8769 

8255  7796 

59 

1  7855  '1.4808 

1.3034 

1.1779 

1.0806  1.0012 

9341 

8760 

8247  7789 

6fl 

1.  7782  !  1.4771 

1.301C 

I  1761 

1.0792  1.0000 

9331 

8751 

8239  7782 

TABLE  XCVI.     Logistical  Logarithms. 


109 


' 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

" 

600 

t,60 

720 

780 

840 

900 

960 

1020 

1080 

1140 

1200 

1260 

0 

7782 

7368 

6990 

6642 

6320 

6021 

5740 

5477 

5229 

4994 

4771 

4559 

1 

7774 

7361 

6984 

6637 

6315 

6016 

5736 

5473 

5225 

4990 

4768 

4556 

2 

7767 

7354 

6978 

6631 

6310 

6011 

5731 

5469 

5221 

4986 

4764 

4552 

3 

7760 

7348 

6972 

6625 

6305 

6006 

5727 

5464 

5217 

4983 

4760 

4549 

4 

7753 

7341 

6966 

6620 

6300 

6001 

5722 

5460 

5213 

4979 

4757 

4546 

5 

7745 

7335 

6960 

6614 

6294 

5997 

5718 

5456 

5209 

4975 

4753 

4542 

6 

7738 

7328 

6954 

6609 

6289 

5992 

5713 

5452 

5205 

4971 

4750 

4539 

7 

7731 

7322 

6948 

6603 

6284 

5987 

5709 

5447 

5201 

4967 

4746 

4535 

8 

7724 

731  5  i  6942 

6598 

6279 

5982 

5704 

5443 

5197 

4964 

4742 

4532 

9 

7717 

7309 

6936 

6592 

6274 

5977 

5700 

5439 

5193 

4960 

4739 

4528 

10 

7710 

7002 

6930 

6587 

6269  I  5973 

5695 

5435 

5189 

4956 

4735 

4525 

11 

7703 

/296 

6924 

6581 

6264 

5968 

5691 

5430 

5185 

4952 

4732 

4522 

12 

7696 

7289 

6918 

6576 

6259 

5963 

5686 

5426 

5181 

4949 

4728 

4513 

13 

7688 

7283 

6912 

6570 

6254 

5958 

5682 

5422 

5177 

4945 

4724 

4515 

14 

768i  J7276 

6906 

6565 

6248 

5954 

5677 

5418 

5173 

4941 

4721 

4511 

15 

767't  7270 

6900 

6559 

6243 

5949 

5673 

5414 

5169 

4937 

4717 

4508 

16 

7667 

'.''264 

6894 

6554 

6238 

5944 

5669 

5409 

5165 

4933 

4714 

4505 

17 

7660 

7257 

6888 

6548 

6233 

5939 

5664 

5405 

5161 

4930 

4710 

4501 

18 

7653 

7251 

6882 

6543 

6228 

5935 

5660 

5401 

5157 

4926 

4707 

4498 

19 

7646 

7244 

6877 

6538 

b'223 

5930 

5655 

5397 

5153 

4922 

4703 

4494 

20 

7639 

7238 

6871 

6532 

6218 

5925 

5651 

5393 

5149 

4918 

4699 

4491 

21 

7632 

7232 

6865 

6527 

6213 

5920 

5646 

5389 

5145 

4915 

4696 

4488 

22 

7625 

7225 

6859 

6521 

6208 

5916 

5642 

5384 

5141 

4911 

4692 

4484 

23 

7618 

7219 

6853 

6516 

6203 

5911 

5637 

5380 

5137 

4907 

4689 

4481 

24|7611 

7212 

6847 

6510 

6198 

5906 

5633 

5376 

5133 

4903 

4685 

4477 

25  7604 

7206 

6841 

6505 

6193 

5902 

5629 

5372 

5129 

4900 

4682 

4474 

26  7597 

7200 

6836 

6500 

6188 

5897 

5624 

5368 

5125 

4896 

4678 

4471 

27  !  7590 

7193 

6830 

6494 

6183 

5892 

5620 

5364 

5122  4892 

4675 

4467 

28  !  7383 

7187 

6824 

6489 

6178 

5888 

5615 

5359 

5118  4889 

4671 

4464 

29  7577 

7181 

6818 

6484 

6173 

5883 

5611 

5355 

511414885 

4668 

4460 

30  7570 

7175 

6812 

6478 

6168 

5878 

5607 

5351 

5110  4881 

4664 

4457 

31  7563 

7168 

6807 

6473 

6163 

5874 

5602 

5347 

5106  4877 

4660 

4454 

32  7556 

7162 

6801 

6467 

6158 

5869 

5598 

5343 

5102  4874 

4657 

4450 

33  7549 

7156 

6795 

6462 

6153 

5864 

5594 

5339 

5098  4870 

4653 

4447 

34  7542 

7149 

6789 

6457 

6148 

5860 

5589 

5335 

5094  4866 

4650 

4444 

35  7535 

7143 

6784  6451 

6143 

5855 

5585 

5331 

5090  4863 

4646 

4440 

36  7528 

7137 

6778 

6446 

6138 

5850 

5580 

5326 

5086 

4859 

4643 

4437 

37  7522 

7131 

6772 

6441 

6133 

5846 

5576 

5322 

5082 

4855 

4639 

4434 

38  7515 

7124 

6766 

6435 

6128 

5841 

5572 

5318 

5079  4852 

4636 

4430 

39  7508 

7118 

6761 

6430 

6123 

5836 

5567 

5314 

5075  '  4848 

4632 

4427 

40  7501 

7112 

6755 

6425  6118 

5832 

5563 

5310 

5071 

4844 

4629 

4424 

41  !  7494 

7106 

6749 

6420  6113 

5827 

5559 

5306 

5067 

4841 

4625 

4420 

42  !  7488 

7100 

6743 

6414  6108 

5823 

5554 

5302 

5063 

4837 

4622 

4417 

43  7481 

7093 

6738 

6409 

6103 

5818 

5550 

5298 

5059 

4833 

4618 

4414 

44  7474 

7087 

6732 

6404 

6099 

5813 

5546 

5294 

5055 

4830 

4615 

4410 

45  7467 

7081 

6726 

6398 

6094 

5809 

5541 

5290 

5051 

4826 

4611 

4407 

46 

7461 

7075 

6721 

6393 

6089 

5804 

5537 

'5285 

5048 

4822 

4608 

4404 

47 

7454 

7069 

6715 

6388 

6084 

5800 

5533 

5281 

5044 

4819  1  4604 

4400 

48 

7447 

7063 

6709 

6383 

6079 

5795 

5528 

5277 

5040  4815 

4601 

4397 

49 

7441 

7057 

6704 

6377 

6074 

5790 

5524 

5273 

5036)4811 

4597 

4394 

50 

7434 

7050 

6698 

6372 

6069 

5786 

5520 

5269 

5032 

4808 

4594 

4390 

51 

7427 

7044 

6692 

6367 

6064 

5781 

5516 

5265 

5028 

4804 

4590 

4387 

52 

7421 

7038 

6687 

6362 

6059 

5777 

5511 

5261 

5025  :  4800 

4587 

4384 

53 

7414 

7032 

6681 

6357 

6055 

5772 

5507 

5257 

5021 

4797 

4584 

4380 

54 

7407 

7026 

6676 

6351 

6050  !  5768 

5503 

5253  5017 

4793 

4580 

4377 

55 

7401 

7020 

6670 

6346 

6045  I  5763 

5498 

5249 

5013 

4789  '4577 

4374 

56 

7394 

7014 

6664 

6341 

6040 

5758 

5494 

5245 

5009 

4786  ,  4573 

4370 

57 

7387 

7008 

6659 

6336 

6035 

5754 

5490 

5241 

5005 

47S2  4570 

4367 

58 
59 

7:381 
7374 

7002 
6996 

6653 
6648 

6331 
6325 

6030  j  5749 
6025  1  5745 

5486 
5481 

5237 
5233 

5002 
4998 

4778 
4775 

4506 
4563 

4364 
4361 

60 

7368 

6990  b«42 

6320 

6021  1  5740 

5477 

5229 

4994 

4771  '4559  4357 

no 


TABLE  XCVI. 


Logistical  Logarithms. 


22   23 

24 

25 

2(3 

27 

28 

29 

30 

31   32 

33  ^ 

'  f 

1320,1380 

1440 

1500 

1560 

1630 

1680 

1740 

1800 

1860 

1920 

1J80 

0 

4357  i  4164 

3979 

3802 

3632 

3468 

3310 

3158 

3010 

2868 

2730 

2596 

1 

4354 

4161 

3976 

3799 

3629 

3465 

3307 

3155 

30Q8 

2866 

2728 

2594 

2 

4351 

4158 

3973 

3796 

3626 

3463 

3305 

3153 

3005 

2863 

2725 

2592 

3 

4347 

4155 

3970 

3793 

3623 

3460 

3302 

3150 

3003 

2861 

2723 

2590 

4 

4344 

4152 

3967 

3791 

3621 

3457 

3300 

3148 

3001 

2859 

2721 

2588 

5 

4341  I  4149 

3964 

3788 

3618 

3454 

3297 

3145 

2998 

2856 

2719 

2585 

6 

4338  1  4145 

3961 

3785 

3615 

3452 

3294 

3143 

2996 

2854 

2716 

2583 

7 

4334 

4142 

3958 

3782  3612 

3449 

3292 

3140 

2993 

2852 

2714 

2581 

8 

4331 

4139 

3955 

3779  3610 

3446 

3289 

3138 

2991 

28-'  9 

2712 

2579 

9 

4328 

4136 

3952 

3776  3607 

3444 

3287 

3135 

2989 

284/1  2710 

2577 

10 

4325 

4133 

3949 

3773  3604 

3441 

3284 

3133 

2986 

2845 

3707 

2574 

11 

4321 

4130 

3946 

3770  1  3601 

3438 

3282 

3130 

2984 

2842 

2705 

2572 

12 

4318 

4127 

3943 

3768 

3598 

3436 

3279 

3128 

2981 

2840 

2703 

2570 

13 

4315 

4124 

3940 

3765 

3596 

3433 

3276 

3125 

2979 

2838 

2701 

2568 

14 

4311 

4120 

3937 

3762 

3593 

3431 

3274 

3123 

2977 

2835 

2698 

2566 

15 

4308 

4117 

3934 

3759 

3590 

3428 

3271 

3120 

2974 

2833 

2696 

2564 

16 

4305 

4114 

3931 

3756 

3587 

3425 

3269 

3118 

2972 

2831 

2694 

2561 

17 

4302 

4111 

3928 

3753 

3585 

3423 

3266 

3115 

2969 

2828 

2692 

2559 

18 

4298 

4108 

3925 

3750 

3582 

3420 

3264 

3113 

2967 

2826 

2689 

2557 

19 

4295 

4105 

3922 

3747 

3579 

3417 

3261 

3110 

2965 

2824 

2687 

2555 

20 

4292 

4102 

3919 

3745 

3576 

341£ 

3259 

3108 

2962 

2821 

2685 

2553' 

21 

4289 

4099 

3917 

3742 

3574 

3412 

3256 

3105 

2960 

2819 

2683 

2551 

22 

4285 

4096 

3914 

3739 

3571 

3409 

3253 

3103 

2958 

2817 

2681 

2548 

23 

4282 

4092 

3911 

3736 

3568 

3407 

3251 

3101 

2955 

2815 

2678 

2546 

24 

4279 

4089 

3908 

3733 

3565 

3404 

3248 

3098 

2953 

2812 

2676 

2544 

25 

4276 

4086 

3905 

3730 

3563 

3401 

3246 

3096 

2950 

2810 

2674 

2542 

26 

4273 

4083 

3902 

3727 

3560 

3399 

3243 

3093 

2948 

2808 

2672 

2540 

27 

4269 

4080 

3899 

3725 

3557 

3396 

3241 

3091 

2946 

2805 

2669 

2538 

28  4266 

4077 

3896 

3722 

3555 

3393 

3238 

3088 

2943 

2803 

2667 

2535 

29  4263 

4074 

3893 

3719 

3552 

3391 

3236 

3086 

2941 

2801 

2665 

2533 

30 

4260 

4071 

3890 

3716 

3549 

3388 

3233 

30S3 

2939 

2798 

2663 

2531 

31 

4256 

4068 

3887 

3713 

3546 

3386 

3231 

3081 

2936 

2796 

2660 

2529 

32 

4253 

4065 

3884 

3710 

3544 

3383 

3228 

3078 

2934 

2794 

2658 

2527 

33 

4250 

4062 

3881 

3708 

3541 

3380 

3225 

3076 

2931 

2792 

2656 

2525 

34 

4247 

4059 

3878 

3705 

3538 

3378 

3223 

3073 

2929 

2789 

2654 

2522 

35 

4244 

4055 

3875 

3702 

3535 

3375 

3220 

3071 

2927 

2787 

2652 

2520 

36 

4240 

4052 

3872 

3699 

3533 

3372 

3218 

3069 

2924 

2785 

2649 

2518 

37 

4237 

4049 

3869 

3696 

3530 

3370 

3215 

3066 

2922 

2782 

2647 

2516 

38 

4234 

4046 

3866 

3693 

3527 

3367 

3213 

3064 

2920 

2780 

2645 

2514 

39 

4231 

4043 

3863 

3691 

3525 

3365 

3210 

3061 

2917 

2778 

2643 

2512 

40 

4228 

4040 

3860 

3688 

3522 

3362 

3208 

3059 

2915 

2775 

2640 

2510 

41 

4224 

4037 

3857 

3685 

3519 

3359 

3205 

3056 

2912 

2773 

2638 

2507 

42 

4221 

4034 

3855 

3682 

3516 

3357 

3203 

3054 

2910 

2771 

2636 

2505 

43 

4218 

4031 

3852 

3679 

3514 

3354 

3200 

3052 

2908 

2769 

2634 

2503 

44 

4215 

4028 

3849 

3677 

3511 

3351 

3198 

3049 

2905 

2766 

2632 

2501 

45 

4212 

4025 

3846 

3674 

3508 

3349 

3195 

3047 

2903 

2764 

2629 

2499 

46 

4209 

4022 

3843 

3671 

3506 

3346 

3193 

3044 

2901 

2762 

2627 

2497 

47 

4205 

4019 

3840 

3668 

3503 

3344 

3190 

3042 

2898 

2760 

2625 

2494 

48 

4202 

4016 

3837 

3665 

3500 

3341 

3188 

3039 

2896 

2757 

2623 

2492 

49 

4199 

4013 

3834 

3663 

3497 

3338 

3185 

3037 

2894 

2755 

2621 

2490 

50 

4196 

4010 

3831 

3660 

3495 

3336 

3183 

3034 

2891 

2753 

2618 

2488 

51 

4193 

4007 

3828 

3657 

3492 

3333 

3180 

3032 

2889 

2750 

2616 

2486 

52 

4189 

4004 

3825 

3654 

3489 

3331 

3178 

3030 

2887 

2748 

2614 

2484 

53 

4186 

4001 

3822  3651 

3487 

3328 

3175 

3027 

2884 

2746 

2612 

2482 

54 

4183 

3998  3820  3649  3484 

3325 

3173 

3025 

2882 

2744 

2610 

2480 

55 

4180 

3995 

3817 

3646 

3481 

3323 

3170 

3022 

2880 

2741 

2607 

2477 

56 

4177 

3991 

3814 

36*3 

3479 

3320 

3168 

3020 

2877 

2739 

2605 

2475 

57 

4174 

3988 

3811 

3640 

3476 

3318 

3165 

3018 

2875 

2737 

2603 

2473 

58 

4171 

3985 

3808 

3637 

3473 

3315 

3163 

3015 

2873 

2735 

2601 

2471 

59 

4167 

3982 

3805 

3635 

3471 

3313 

3160 

3013 

2870 

2732 

2599 

2469 

60 

4164 

3979 

3802 

3632  3468 

3310 

3158 

3010 

2868 

2730 

2596 

2467 

TABLE  XCVI.     Logistical  Logarithms. 


Ill 


' 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

2040 

2100 

2160  2220 

2280 

2340 

2400 

2460 

2520 

2580 

2640 

2700 

0 

2467 

2341 

2218  2099 

1984 

1871 

1761 

1654 

1549 

1447 

1347 

1249 

1 

2465 

2339 

2216 

2098 

1982 

1869 

1759 

1652 

1547 

1445 

1345 

1248 

2 

2462 

2337 

2214 

2096 

1980 

1867 

1757 

1650 

1546 

1443 

1344 

1246 

3 

2460 

2335 

2212 

2094 

1978 

1865 

1755 

1648 

1544 

1442 

1342 

1245 

4 

2458 

2333 

2210 

2092 

1976 

1863 

1754 

1647 

1542 

1440 

1340 

1243 

5 

2456 

2331 

2208 

2090 

1974 

1862 

17*2 

1645 

1540 

1438 

1339 

1241 

6 

2454 

2328 

2206 

2088 

1972 

1860 

1750 

1643 

1539 

1437 

1337 

1240 

7 

2452 

2326 

2204 

2086 

1970 

1858 

1748 

1641 

1537 

1435 

1335 

1238 

8 

2450 

2324 

2202 

2084 

1968 

1856 

1746 

1640 

1535 

1433 

1334 

1237 

9 

2448 

2322 

2200 

2082 

1967 

1854 

1745 

1638 

1534 

1432 

1332 

1235 

10 

2445 

2320 

2198 

2080 

1965 

1852 

1743 

1636 

1532 

1430 

1331 

1233 

11 

2443 

2318 

2196 

2078 

1963 

1850 

1741 

1634 

1530 

1428 

1329 

1232 

12 

2441 

2316 

2194 

2076 

1961 

1849 

1739 

1633 

1528 

1427 

1327 

1230 

13 

2439 

2314 

2192 

2074 

1959 

1847 

1737 

1631 

1527 

1425 

1326 

1229 

14 

2437 

2312 

2190 

2072 

1957 

1845 

1736 

1629 

1525 

1423 

1324 

1227 

15 

2435 

2310 

2188 

2070 

1955 

1843 

1734 

1627 

1523 

1422 

1322 

1225 

16 

2433 

2308 

2186 

2068 

1953 

1841 

1732 

1626 

1522 

1420 

1321 

1224 

17 

2431 

2306 

2184 

2066 

1951 

1839 

1730 

1624 

1520 

1418 

1319 

1222 

18 

2429 

2304 

2182 

2064 

1950 

1838 

1728 

1622 

1518 

1417 

1317 

1221 

19 

2426 

2302 

2180 

2062 

1948 

1836 

1727 

1620 

1516 

1415 

1316 

1219 

20 

2424 

2300 

2178 

2061 

1946 

1834 

1725 

1619 

1515 

1413 

1314 

1217 

21 

2422 

2298 

2176 

2059 

1944 

1832 

1723 

1617 

1513 

1412 

1313 

1216 

22 

2420 

2296 

2174 

2057 

1942 

1830 

1721 

1615 

1511 

1410 

1311 

1214 

23 

2418 

2294 

2172 

2055 

1940 

1828 

1719 

1613 

1510 

1408 

1309 

1213 

24 

2416 

2291 

2170 

2053 

1938 

1827 

1718 

1612 

1508 

1407 

1308 

1211 

25 

2414 

2289 

2169 

2051 

1936 

1825 

1716 

1610 

1506 

1405 

1306 

1209 

26 

2412 

2287 

2167 

2049 

1934 

1823 

1714 

1608 

1504 

1403 

1304 

1208 

27 

2410 

2285* 

2165 

2047 

1933 

1821 

1712 

1606 

1503 

1402 

1303 

1206 

28 

2408 

2283 

2163 

2045 

1931 

1819 

1711 

1605 

1501 

1400 

1301 

1205 

29 

2405 

2281 

2161 

2043 

1929 

1817 

1709 

1603 

1499 

1398 

1300 

1203 

30 

2403 

2279 

2159 

2041 

1927 

1816 

1707 

1601 

1498 

1397 

1298 

1201 

31 

2401 

2277 

2157 

2039 

1925 

1814 

1705 

1599 

1496 

1395 

1296 

1200 

32 

2399 

2275 

2155 

2037 

1923 

1812 

1703 

1598 

1494 

1393 

1295 

1198 

33 

2397 

2273 

2153 

2035 

1921 

1810 

1702 

1596 

1493 

1392 

1293 

1197 

34 

2395 

2271 

2151 

2033 

1919 

1808 

1700 

1594 

1491 

1390 

1291 

1195 

35 

2393 

2269 

2149 

2032 

1918 

1806 

1698 

1592 

1489 

1388 

1290 

1193 

36 

2391 

2267 

2147 

2030 

1916 

1805 

1696 

1591 

1487 

1387 

1288 

1192 

37 

2389 

2265 

2145 

2028 

1914 

1803 

1694 

1589 

1486 

1385 

1287 

1190 

38 

2387 

2263 

2143 

2026 

1912 

1801 

1693 

1587 

1484 

1383 

1285 

1189 

39 

2384 

2261 

2141 

2024 

1910 

1799 

1691 

1585 

1482 

1382 

1283 

1187 

40 

2382 

2259  2139 

2022 

1908 

1797 

1C89 

1584 

1481 

1380 

1282 

1186 

41 

2380 

2257 

2137 

2020 

1906 

1795 

1687 

1582 

1479 

1378 

1280 

1184 

42 

2378 

2255 

2135 

2018 

1904 

1794 

1686 

1580 

1477 

1377 

1278 

1182 

43 

2376 

2253 

2133 

2016 

1903 

1792 

1684 

1578 

1476 

1375 

1277 

1181 

44 

2374 

2251 

2131 

2014 

1901 

1790 

1682 

1577 

1474 

1373 

1275  1179 

45 

2372 

2249 

2129 

2012 

1899 

1788 

1680 

1575 

1472 

1372 

1274  1178 

46 

2370 

2247 

2127 

2010 

1897 

1786 

1678 

1573 

1470 

1370 

1272  1176 

47 

2368 

2245 

2125 

2009 

1895 

1785 

1677 

1571 

1469 

1368 

1270 

1174 

48 

2366 

2243 

2123 

2007 

1893 

1783 

1675 

1570 

1467 

1367 

1269 

1173 

49 

2364 

2241 

2121 

2005 

1891 

1781 

1673 

1568 

1465 

1365 

1267!  11  71 

50 

2362 

2239 

2119 

2003 

1889 

1779 

1671 

1566 

1464 

1363 

1266  1170 

51 

2359 

2237 

2117 

2001 

1888 

1777 

1670 

1565 

1462 

1362 

1264  1168 

52 

2357 

2235 

2115 

1999 

1886 

1775 

1668 

1563 

1460 

1360 

1262  !  1167 

53 

2355 

2233 

2113 

1997 

1884 

1774 

1666 

1561 

1459 

1359 

1261  1165 

54 

2353 

2231 

2111 

1995 

1882 

1772 

1664 

1559 

1457  I  1357 

1259  I  1163 

55 

2351 

2229 

2109 

1993 

1880 

1770 

1663 

1558  1455 

1355 

1257 

1162 

56 

2349  |  2227 

2107 

1991 

1878 

1768 

1661 

1556 

1454 

1354 

1256 

1160 

57 

2347 

2225 

2105 

1989 

1876 

1766 

1659 

1554 

1452 

1352 

1254 

1159 

58 

2-345 

2223 

2103 

1987 

1875 

1765 

1657 

1552 

1450 

1350 

1253 

1157 

59 

2343 

2220 

2101 

1986 

1873 

1763 

1655 

1551 

1449 

1349 

1251 

1156 

60 

2341 

2218 

2099 

1984 

1871 

1761 

1654 

1549 

1447 

1347 

1249 

1154 

112 


TABLE  XCVI.     Logistical  Logarithms. 


r  — 

1 

46 

47 

48 

49 

50 

51  |  52 

53 

54 

55 

56 

57 

58 

59 

•" 

2760 

2820 

2880 

2940 

3000 

3060  3120 

3180 

3240 

3300 

3360 

3420 

3480 

3540 

~0 

1154 

1061 

"0969 

"0880 

"0792"  i  0706 

0621 

0539 

0458 

0378 

0300 

0223 

0147 

0073 

1 

1152 

1059 

0968 

0878 

0790  i  0704 

0620 

0537 

0456 

0377 

0298 

0221 

0145 

0072 

2 

1151 

1057 

0966 

OS77 

0789 

0703 

0619 

0536 

0455 

0375 

0297 

0220 

0145 

0071 

3 

1149 

1056 

0965 

0875 

0787 

0702 

0617 

0535 

0454 

0374 

0296 

0219 

0143 

0069 

4 

1148 

1054 

0963 

0874 

0786 

0700 

0616 

0533 

0452 

0373 

0294 

0218 

0142 

0068 

5 

1146 

1053 

0962 

0872 

0785 

0699 

0615 

0532 

0451 

0371 

0293 

0216 

0141 

0067 

6 

1145 

1051 

0960 

0871 

0783 

0697 

0613 

0531 

0450 

0370 

0292 

0215 

0140 

0066 

7 

1143 

1050 

0859 

0869 

0782 

0696 

0612 

0529 

0448 

0369 

0291 

0214 

0139 

0064  i 

8 

1141 

1048 

0957 

0868 

0780 

0694 

0610 

0528 

0447 

0367 

0289 

0213 

0137 

0063 

9 

1140 

1047 

0956 

0866 

0779 

0693 

0609 

0526 

0446 

0366 

0288 

0211 

0136 

0062 

10 

1138 

1045 

0954 

0865 

0777 

0692 

0608 

0525 

0444 

0365 

0287 

0210 

0135 

0061 

11 

1137 

1044 

0953 

0863 

0776 

0690 

0606 

0524 

0443 

0363 

0285 

0209 

0134 

0060 

12 

1135 

1042 

0951 

0862 

0774 

0689 

0605 

0522 

0442 

0362 

0284 

0208 

0132 

0058 

13 

1134 

1041 

0950 

0860 

0773 

0687 

0603 

0521 

0440 

0361 

0283 

0206 

0131 

0057 

14 

1132 

1039 

0948 

0859 

0772 

0686 

0602 

0520 

0439 

0359 

0282 

0205 

0130 

0056 

15 

1130 

1037 

0947 

0857 

0770 

0685 

0601 

0518 

0438 

0358 

0280 

0204 

0129 

0055 

16 

1129 

1036 

0945 

0856 

0769 

0683 

0599 

0517 

0436 

0357 

0279 

0202 

0127 

0053 

17 

1127 

1034 

0944 

0855 

0767 

0682 

0598 

0516 

0435 

0356 

0278 

0201 

0126 

0052 

|18 

1126 

1033 

0942 

0853 

0766 

0680 

0596 

0514 

0434 

0354 

0276 

0200 

0125 

0051 

19 

1124 

1031 

0941 

0852 

0764 

0679 

0595 

0513 

0432 

0353 

0275 

0199 

0124 

0050 

20 

1123 

1030 

0939 

0850 

0763 

0678 

0594 

0512 

0431 

0352 

0274 

0197 

0122 

0049 

21 

1121 

1028 

0938 

0849  0762 

0676 

0592 

0510 

0430 

0350 

0273 

0196 

0121 

0047 

22 

1119 

1027 

0936 

0847  0760 

0675 

0591 

0509 

0428 

0349 

0271 

0195 

0120 

0046 

23 

1118 

1025 

0935 

0846  0759 

0673 

0590 

0507 

0427 

0348 

0270 

0194 

0119 

0045 

24 

1116 

1024 

0933 

0844  0757 

0672 

0588 

0506 

0426 

0346 

0269 

0192 

0117 

0044 

25 

1115 

1022 

0932 

0843  0756 

0670  0587 

0505 

0424 

0345 

0267 

0191 

0116 

0042 

26 

1113 

1021 

0930 

0841  0754 

0669  0585 

0503 

0423 

0344 

0266 

0190 

0115 

0041 

27 

1112 

1019 

0929 

0840  0753 

0668  0584 

0502 

0422 

0342 

0265 

0189 

0114 

0040 

28 

1110 

1018 

0927 

0838 

0751 

0666 

0583 

0501 

0420 

0341 

0264 

0187 

0112 

0039 

29 

1109 

1016 

0926 

0837 

0750 

0665 

0581 

0499 

0419 

0340 

0262 

0186 

0111 

0038 

30 

1107 

1015 

0924 

0835 

0749 

0663 

0580 

0498 

0418 

0339 

0261 

0185 

0110)0036 

31 

1105 

1013 

0923 

0834  0747 

0662 

0579 

0497 

0416 

0337 

0260 

0184 

0109 

0035 

32 

1104 

1012 

0921 

0833  0746 

0661 

0577 

0495 

0415 

0336 

0258 

0182 

0107 

0034 

33 

1102 

1010 

0920 

0831 

0744 

0659 

0576 

0494 

0414 

0335 

0257 

0181 

0106 

0033  | 

34 

1101 

1008 

0918 

0830 

0743 

0658  0574 

0493 

0412 

0333 

0256 

0180 

0105 

0031 

35 

1099 

1007 

0917 

0828  i  0741 

0656  0573 

0491 

0411 

0332 

0255 

0179 

0104 

0030 

36 

1098 

1005 

0915 

0827  0740 

0655  0572 

0490 

0410 

0331 

0253 

0177 

0103 

0029 

37 

1096 

1004 

0914 

0825  0739 

0654  0570 

0489 

0408 

0329 

0252 

0176 

0101 

0028 

38 

1095 

1002 

0912 

0824  ;  0737 

0652 

0569 

0487 

0407 

032S 

0251 

0175 

0100 

0027 

39 

1093 

1001 

0911 

0822  0736 

0651 

0568 

0486 

0406 

0327 

0250 

0174 

0099 

0025 

40 

1091 

0999 

0909 

0821  0734 

0649 

0566 

0484 

0404 

0326 

0248 

0172 

0098 

0024 

41 

1090 

0998 

0908 

0819  0733 

0648  0565 

0483 

0403 

0324 

0247 

0171 

0096 

0023 

42 

1088 

0996 

0906 

0818,0731 

0647  i  0563 

0482 

0402 

0323 

0246 

0170 

0095 

0022 

43 

1087 

0995 

0905 

0816!  0730 

0645  !  0562 

0480 

0400 

0322 

0244 

0169 

0094 

0021 

44 

1085 

0993 

0903 

0815 

0729 

0644  0561 

0479 

0399 

0320 

0243 

0167 

0093 

0019 

45 

1084 

0992 

0902 

0814 

0727 

0642  0559 

0478 

0398 

0319 

0242 

0166 

009110018 

46 

1082 

0990 

0900 

6812 

0726 

0641  0558 

0476 

0396 

0318 

0241 

0165 

0090  0017 

47 

1081 

0989 

0899 

0811 

0724 

0640  0557 

0475 

0395 

0316 

0239 

0163 

0089  0016 

48 

1079 

0987 

0897 

0809 

0723 

0638  |  0555 

0474 

0394 

0315 

0238 

0162 

OQ88 

0015 

49 

1078 

0986 

0896 

0808  0721 

0637  0554 

0472  -  0392 

0314 

0237 

0161 

0087 

0013 

60 

1076 

0984 

0894 

0806 

0720 

0635 

0552 

0471 

0391 

031310235 

0160 

0085 

0012 

51 

1074 

0983 

0893 

0805 

0719 

0634 

0551 

0470 

0390 

0311  0234 

0158 

0084  0011 

52 

1073 

0981 

0891 

0803 

0717 

0633 

0550 

0468 

0388 

0310 

0233 

0157 

0083  0010 

53 

1071 

0980 

0890 

0802  0716 

0631 

0548 

0467 

0387 

0309 

0232 

0156 

0082  0008 

54 

1070 

0978 

0888 

0801  !  0714 

0630 

0547 

0466 

0386 

0307 

0230 

0155 

0080(0007 

55 

1068 

0977  ,  0887 

0799  |  0713 

0628 

0546 

0464 

0384  0306  j  0229  j  0153 

0079 

0006 

66 

1067 

0975 

0885  0798  0711 

3627  0544 

0463 

0383  :  0305  i  09C8  0152 

0078 

0005 

fi7 

1065 

0974 

0884 

0796 

0710 

0626  0543 

0462 

0382  0304-  0227 

0151 

0077 

0004 

98 

1064 

0972 

0883 

0795 

0709 

0624  !  0541  1  0460 

0381 

0302  !  0225 

0150 

0075  0002 

r>9 

1062 

0971 

0881 

0793 

0707 

0623  0540 

0459 

0379 

0301 

0224 

0148 

0074  0001 

no 

1061 

0969 

0880 

0792 

0706 

0621  i0539 

0458 

0378 

0300 

0223 

0147 

0073  0000 

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